UNIVERSITE DU QUEBEC A MONTREAL

Remplissages d’une composante du bord del’exterieur de l’entrelacs de Whitehead

THESE

PRESENTEE

COMME EXIGENCE PARTIELLE

DU DOCTORAT EN MATHEMATIQUE

PAR

GABRIEL INDURSKIS

DECEMBRE 2005

UNIVERSITE DU QUEBEC A MONTREAL

Fillings of one boundary component of theWhitehead link exterior

PH.D. THESIS

PRESENTED

AS A PARTIAL REQUIREMENT

FOR THE DOCTORATE IN MATHEMATICS

BY

GABRIEL INDURSKIS

DECEMBER 2005

Acknowledgements

Many people have contributed to help me reach this point, and they all deservemy heartfelt thanks.

First of all, I would like to thank my supervisor, Steven Boyer. His endlesspatience and confidence in me and my abilities (in particular at times when I didn’thave confidence in either) had no little part in making me carry on past the manyrocks in the road. I couldn’t have done it without his never-waning support — be itmathematical, psychological, or financial. Thank you!

The next person I will certainly need to mention here is Stephan Tillmann. Apartfrom being a great inspiration mathematically (thanks for suggesting the detour!) and agreat person to talk (or listen) to, he also happened to become one of my closest friendsover the last years — not least by making sure I was well-fed during the more strenuousparts of the writing stage. Danke, Tillus!

I also would like to extend my thanks to everybody at the mathematics depart-ment at UQAM, who have made my stay such a pleasurable time. First, to all membersof CIRGET, in particular to Olivier Collin, Vestislav Apostolov and Andre Joyal, formany interesting conversations — and not least for all the good times spent in the CIR-GET coffee room. Second, to Alexandra Haedrich, who never failed to help with anybureaucratic or non-bureaucratic issue at hand, and who spread a warmth throughoutthe department with her infectious smile and good nature. Also, to the administrativestaff of the department: Jeanne Jobin, Rita Chamberland, Marie-Claude Cote, and inparticular to Manon Gauthier, who all made sure that every trip to the office was a plea-surable one and I never had to worry too much about bureaucratic details. Finally, toGisele Legault, for keeping the UNIX servers running smoothly, and to Andre Lauzon,for never-ending technical support and discussions. Merci beaucoup!

Thanks must also go to the Institut de sciences mathematiques for the generousfinancial support. Merci beaucoup!

I also would like to thank my mom, who — even after many arduous explanationattempts on my part — probably still doesn’t quite know what it is I’m doing, but whonevertheless never stops to support and believe in me. Vielen, vielen Dank!

Finally, I need to thank all my wonderful friends, in Montreal and elsewhere in theworld, for simply being the wonderful people you are. I couldn’t have done it withoutyou around me (even if you were actually physically far away). I won’t try to list youall — you know who you are.

Merci — Thanks — Danke — Gracie!

Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1 Some algebraic varieties associated to 3-manifolds . . . . . . . . . . . . . . 8

1.2 Culler-Shalen seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Elimination theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 The Whitehead link exterior . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 2The set of p-reps of a once-filled Whitehead link exterior . . 29

2.1 Detour through the eigenvalue variety . . . . . . . . . . . . . . . . . . . . . 32

2.2 The subset of the eigenvalue variety corresponding to p-reps . . . . . . . . . 34

2.3 Solutions corresponding to reducible representations . . . . . . . . . . . . . 35

2.4 Solutions corresponding to irreducible representations . . . . . . . . . . . . 36

2.5 The number of conjugacy classes of p-reps . . . . . . . . . . . . . . . . . . . 41

2.6 The eigenvalue resultant and its properties . . . . . . . . . . . . . . . . . . 43

Chapter 3The minimal non-zero value of the total Culler-Shalen semi-norm and p-reps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1 Smoothness of the character variety at the characters of p-reps . . . . . . . 60

3.2 Deformations of representations and degree of the trace function . . . . . . 75

iii

Chapter 4The total Culler-Shalen seminorm . . . . . . . . . . . . . . . . . . . . 80

4.1 Boundary slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 The Culler-Shalen seminorm of the Seifert fillings . . . . . . . . . . . . . . . 88

4.3 The total Culler-Shalen seminorm of the filled manifolds . . . . . . . . . . . 100

Chapter 5Fillings of integer fillings of the Whitehead link exterior . . 112

5.1 Once-punctured torus bundles and their monodromies . . . . . . . . . . . . 113

5.2 Integer fillings of the Whitehead link exterior and their monodromies . . . . 119

5.3 Closed manifolds obtained by Dehn filling the integer fillings . . . . . . . . 123

5.4 Applying a result by Masters . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

List of Tables

2.1 Number of non–zero solutions (including multiplicities) of the resultant 40

2.2 Upper bound for the number of distinct roots s 6∈ {0,±1} of resp,q . . . 40

2.3 Upper bound for the number of elements in (P′′)i . . . . . . . . . . . . . 42

2.4 Upper bound for the number of elements in P′′ . . . . . . . . . . . . . . 42

2.5 Upper bound for the number of conjugacy classes of SL2(C) p-reps of

π1(Wp/q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Upper bound for the minimal Culler-Shalen seminorm s = ‖µ1‖ for

Wrh(p/q, · ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Boundary curve data for minimal representatives of some vertex solutions

(from [Til02, Table 5.6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 Linear combinations of vertex solutions with integers a, b ≥ 0 and their

boundary curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Variable boundary slopes of Wrh and their ranges . . . . . . . . . . . . . 86

4.4 Possible boundary slopes for Wrh(p/q) (q > 0) . . . . . . . . . . . . . . . 87

4.5 Number of PSL2(C)-characters of π1Wrh(p/q, 1) . . . . . . . . . . . . . . 92

4.6 Number of PSL2(C)-characters of π1Wrh(p/q, 2) . . . . . . . . . . . . . . 93

4.7 Number of PSL2(C)-characters of π1Wrh(p/q, 3) . . . . . . . . . . . . . . 94

4.8 Number of SL2(C) and PSL2(C) characters of the base orbifold groups . 98

v

4.9 Number of non-abelian SL2(C) characters for Seifert fillings of Wrh(p/q)

when p is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.10 Distances between slopes in Wrh(p/q) for p/q ∈ (−∞, 0), p odd, q > 0 . 101

4.11 Distances between slopes in Wrh(p/q) for p/q ∈ (0, 2), p odd . . . . . . . 105

4.12 Distances between slopes in Wrh(p/q) for p/q ∈ (2, 4), p odd . . . . . . . 106

4.13 Distances between slopes in Wrh(p/q) for p/q ∈ (4,∞), p odd . . . . . . 108

4.14 The parameters for the total Culler-Shalen seminorm of Wrh(p/q) for p

odd, q > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Various decompositions of the monodromy of RSSm . . . . . . . . . . . 120

List of Figures

1.1 The right-handed Whitehead link . . . . . . . . . . . . . . . . . . . . . . 24

2.1 The generic distribution of the non-zero roots of resp,q . . . . . . . . . . 48

2.2 The distribution of the non-zero roots of res65,16 . . . . . . . . . . . . . 48

3.1 The roots of the polynomial d2 . . . . . . . . . . . . . . . . . . . . . . . 79

5.1 Conventions for the once-punctured torus bundle Mh . . . . . . . . . . . 114

5.2 The bad slopes for m ≤ −3 odd, 7 - 2−m . . . . . . . . . . . . . . . . . 128

5.3 The bad slopes for m < 0 and 4 | m . . . . . . . . . . . . . . . . . . . . 128

5.4 The bad slopes for m < 0 when 7 | 2−m . . . . . . . . . . . . . . . . . 129

5.5 The bad slopes for m ≥ 7 odd, 7 - m− 2 . . . . . . . . . . . . . . . . . . 129

5.6 The bad slopes for m > 0 when 4 | m− 2 . . . . . . . . . . . . . . . . . 130

5.7 The bad slopes for m > 0 when 7 | m− 2 . . . . . . . . . . . . . . . . . 130

Glossary of Symbols

∼ conjugated

∼= isomorphic or homeomor-

phic

' birational equivalent

.= equality of Laurent poly-

nomials up to multiplica-

tion by units

gcd greatest common divisor

bxc largest integer n ≤ x

N,Z,Q,R,C natural, integer, rational,

real, complex numbers

Z+ non-negative integers (in-

cluding 0)

C∗ C− {0}

Z/p Z/pZ

R SL2(C) representation va-

riety

Rred reducible SL2(C) repre-

sentations

Rirr R−Rred

C a upper/lower triangular

subvariety of R

D a partially diagonal sub-

variety of R

X SL2(C) character variety

Xred reducible SL2(C) charac-

ters

Xirr X− Xred

X0 a non-trivial curve in XeX0 smooth projective model

of X0

E SL2(C) eigenvalue variety

Ered subvariety of E corre-

sponding to reducible rep-

resentations

E0 E− Er

R PSL2(C) representation

variety

X PSL2(C) character vari-

ety

E PSL2(C) eigenvalue vari-

ety

‖α‖X0, ‖α‖ Culler-Shalen seminorm

of slope α on curve X0,

total seminorm

sX0 , s minimal non-zero value of

the Culler-Shalen semi-

norm on X0, total mini-

mal seminorm

resp,q eigenvalue resultant

nresp,q normalized resultant

res′p,q, res′′p,q first and second deriva-

tives

Wlh,Wrh left-/right-handed White-

head link exterior

W Wrh, unless otherwise

noted

Wp/q W (p/q, ·), once-filled

Whitehead link exterior

Tq, Uq q-th Chebyshev polyno-

mial of first and second

type

viii

Resume

Nous etudions des 3-varietesWp/q := W (p/q, · ) obtenues par remplissage de Dehnavec pente p/q sur une composante du bord de l’exterieur de l’entrelacs de WhiteheadW .

En utilisant une parametrisation explicite de la variete des valeurs propres pourW , nous determinons un polynome a une variable dont les racines complexes carac-terisent les p-representations de π1(Wp/q), i.e. les representations avec valeurs dansSL2(C) qui sont paraboliques sur le sous-groupe peripherique. En particulier, nousobtenons une borne superieure pour le nombre de classes de conjugaison de ces represen-tations. Dans la plupart des cas, nous montrons que cette borne est en fait atteinte.

On en deduit une borne superieure pour la valeur minimale non-nulle de la semi-norme de Culler-Shalen de la variete Wp/q. Apres avoir determine les semi-normes ‖α‖pour les pentes α = 1, 2, 3 correspondant a des varietes Wp/q(α) ayant une structurede fibration de Seifert, nous determinons completement la semi-norme de Culler-Shalendans le cas ou p est impaire.

En se limitant aux varietes W (m, · ) ou m ∈ Z, nous etudions les 3-varietes fer-mees W (m, p/q) obtenues par remplissage de Dehn sur la composante du bord restante.Il a ete prouve par Roberts, Shareshian et Stein que pour m ≤ −3 impaire et p ≥ q ≥ 1ces varietes n’admettent pas de feuilletage dont l’espace de feuilles de la feuilletage in-duite sur le revetement universel est homeomorphe a R. De plus, Fenley a montre queces varietes n’admettent pas de laminations essentielles lorsque m ≤ −4 et |p− 2q| = 1.En utilisant des travaux de Masters, nous montrons que ces varietes possedent desrevetements finis avec premier nombre de Betti positif, et donc que ces revetementscontiennent des surfaces essentielles.

Topologie — 3-Varietes — Chirurgie de DehnVariete de Caracteres — Semi-norme de Culler-Shalen

Abstract

We study 3-manifolds Wp/q := W (p/q, · ) obtained by Dehn filling with slope p/qon one boundary component of the Whitehead link exterior W .

Using an explicit parametrization of the eigenvalue variety of W , we determinea one-variable polynomial whose complex roots characterize the so-called p-reps ofπ1(Wp/q), i.e. representations with values in SL2(C) which are parabolic on the periph-eral subgroup. In particular, we obtain an upper bound for the number of conjugacyclasses of these representations, which we show is in fact attained in most cases.

Using this information, we are able to determine an upper bound for the minimalnon-zero Culler-Shalen seminorm of the manifold Wp/q. By explicitly determining theCuller-Shalen seminorm ‖α‖ for the filling slopes α = 1, 2, 3 which are known to yieldSeifert-fibred manifolds Wp/q(α), we completely determine the Culler-Shalen seminormof Wp/q when p is odd.

Restricting attention to the manifolds W (m, · ) where m ∈ Z, we study the closed3-manifolds W (m, p/q) obtained by Dehn filling on the remaining boundary component.It was shown by Roberts, Shareshian and Stein that for m ≤ −3 odd and p ≥ q ≥ 1these manifolds do not admit R-covered foliations, and the manifolds with m ≤ −4 and|p− 2q| = 1 were shown not to admit any essential laminations by Fenley. Using workby Masters, we show that these manifolds have finite covers with positive first Bettinumber, and that these covers hence contain essential surfaces.

Topology — 3-Manifolds — Dehn SurgeryCharacter Variety — Culler-Shalen Seminorm

Introduction

The main focus of this thesis lies in the study of 3-manifolds obtained from another

3-manifoldW by the process of Dehn filling , in which a solid torus V is attached to one of

the torus boundary components of W . The obtained manifold is completely determined

by the isotopy class of a simple closed curve on ∂W , called a slope, which can be

identified1 with a quotient p/q in Q ∪ {∞}. It is a classical result, commonly known

as the fundamental theorem of surgery theory , of Wallace ([Lic62]) and (independently)

Lickorish ([Wal60]) that any closed, connected, orientable 3-manifold can be obtained

by Dehn surgery2 on a framed3 link in S3. This makes Dehn surgery or filling a powerful

tool to obtain new 3-manifolds from a given one. In particular, it is a natural question

to ask which of the properties of the original manifold get transferred onto the newly

obtained one.

One such “transferable” property is the existence of a complete hyperbolic struc-

ture with finite volume. Thurston’s hyperbolic Dehn surgery theorem (cf. [GT87]) shows

that there is at most a finite number of exceptional slopes for which the manifold ob-

tained by Dehn filling from a hyperbolic manifold fails to be hyperbolic. Work of Bleiler

and Hodgson ([BH96]), Agol ([Ago00]) and Lackenby ([Lac00]) then shows that there

1After having chosen a basis for H1(∂W ) = Z⊕ Z.

2The process of first removing an open tubular neighborhood of a orientation-preserving simple

closed curve, and then applying Dehn filling to the new boundary torus.

3A link L = K1 ∪ · · · ∪Kn in S3, each component of which is labeled with an integer slope. The

surgery proceeds along these slopes for each boundary component.

2

are at most 12 fillings which yield non-hyperbolike4 manifolds, and that the distance5

between two such exceptional slopes is at most 10. Given an infinite class of 3-manifolds,

it in general remains a difficult problem to determine the set of exceptional slopes and

the topology of the resulting filled manifolds.

A powerful tool which has enabled tremendous advancement in this respect is the

theory of SL2(C) representations of a 3-manifold group π1(M), pioneered by Culler and

Shalen in [CS83] and since then refined in a long series of papers6. In particular the

so-called Culler-Shalen norm (generalized to seminorms by Boyer and Zhang in [BZ98]),

defined on the Dehn surgery space H1(∂M ; R) ∼= R2, has provided much insight into the

topology of the filled manifolds. Two particular examples of its power are the Cyclic

Dehn Surgery Theorem (cf. [CGLS87]) and the Finite Surgery Theorem (cf. [BZ01]),

which strongly restrict the possibilities for filling slopes which yield a manifold with

cyclic or finite fundamental group, respectively. See [Boy02a] for an excellent survey on

existing results.

One of the major motivations for studying Culler-Shalen seminorms can thus be

seen to lie in determining (bounds for) exceptional fillings. On the other hand, the

direct connection between the Culler-Shalen seminorm and the character variety of a

3-manifold also makes them an important tool to study the structure of this affine

algebraic variety, which is in general hard to completely determine.

It is with this motivation in mind that we study the Culler-Shalen seminorms

of the infinite family of manifolds Wp/q obtained by Dehn filling with slope p/q on

one component of the (right-handed) Whitehead link exterior W . The manifolds Wp/q

4A manifold is hyperbolike if it is irreducible, atoroidal and not Seifert fibered, with infinite word

hyperbolic fundamental group — this is equivalent to the existence of a complete hyperbolic structure

of finite volume if Thurston’s Geometrization Conjecture is true.

5Geometric intersection number

6See in particular [CGLS87,CCG+94,BZ94,BZ96,BZ97,BZ98]. Note this list is not complete

and is only meant to be a starting point for the reader unfamiliar with the subject. See chapter 1 for

more background and references.

3

constitute a rich class of 3-manifolds whose boundary consists of one torus, and contain

important infinite subfamilies of well-known manifolds such as the twist knot exteriors

(for p/q = 1/n, n ∈ Z) and an important class of once-punctured torus bundles (for

p/q = m, m ∈ Z). Note that for example the figure-8 knot complement (p/q = −1)

and its so-called sister manifold (p/q = 5) are part of this family. Since the Whitehead

link exterior admits a complete hyperbolic structure of finite volume, the same is true

for all but a finite number of the manifolds Wp/q. Furthermore, unless p/q ∈ {0, 4},

all manifolds Wp/q are small manifolds, i.e. they do not contain any closed essential

surfaces (cf. [BGZ01, Appendix]).

The main result of this thesis is to completely determine the Culler-Shalen semi-

norm for the manifolds Wp/q when p is odd. This work can be seen as a generalization

of similar work previously done for the twist knots by Boyer, Mattman and Zhang

([BMZ97]) and of the author’s (unpublished) work on once-punctured torus bundles ob-

tained by integer filling on the Whitehead link exterior. The results given in this thesis

completely include these previous results, and extend them to a much larger class of

fillings. In future work, we also plan to include the fillings with p even in our results,

covering all but finitely many of the manifolds Wp/q.

An interesting by-product of this work is the complete determination of the so-

called p-reps of the fundamental group of Wp/q. These are representations into SL2(C)

which are parabolic on the peripheral subgroup7, and were first studied for knot groups

(in particular the groups of 2-bridge knots) by Riley (cf. [Ril72]). In his classical work, he

showed how under certain conditions on the group presentation (e.g. for the fundamental

group of a 2-bridge knot) all such p-reps can be characterized by the complex roots of

a single one-variable polynomial, while in the general situation only a characterization

by a system of polynomial equations can be expected. Passing through the so-called

eigenvalue variety (as defined by Tillmann in [Til05]) of the unfilled Whitehead link

7To be precise, the generator corresponding to the meridian of the boundary torus is sent to

a non-trivial element of trace ±2 in SL2(C). As a consequence, the generator corresponding to the

longitude will be either parabolic as well, or ±I.

4

exterior W , we are able to find a similar approach which seems naturally adapted to

the situation of Dehn filling to determine a one-variable polynomial whose roots are in

one-to-one correspondence with the set of eigenvalues of p-reps. This in turn is shown

to be in bijective correspondence with the set of p-reps itself, and we obtain a complete

characterization of all p-reps of Wp/q. Originally motivated because of its connection

with the minimal non-zero Culler-Shalen seminorm, this result is interesting by itself,

for example because of its connection with the complete hyperbolic structure on the

manifolds Wp/q (if it exists). Such a structure implies the existence of a discrete faithful

representation into SL2(C) which is parabolic on the peripheral subgroup, and which

hence lies in the set of p-reps we determined. In future work, we plan to identify

this representation amongst the set of all p-reps, which then would (amongst other

applications) possibly enable us to determine the trace fields for all hyperbolic fillings

Wp/q.

In a slightly different vein, we study a family of closed manifolds RSSm(p/q) :=

W (m − 2, p/q) obtained by a further Dehn filling of the once-punctured torus bundle

W (m − 2, · ) obtained by integer filling on one boundary component of W . These

manifolds are, apart from a finite number of exceptions, hyperbolic manifolds for |m| > 2

and have appeared repeatedly in connection with some important open questions in 3-

manifold topology. An important result about these manifolds was shown by Roberts,

Shareshian and Stein in [RSS03], where they proved that for m < −2 odd and p ≥ q ≥ 1

with p odd, any action of the fundamental group of RSSm(p/q) on an R-order tree8 is

trivial, i.e. has a global fixed point9. They furthermore show that this implies that

the manifold RSSm(p/q) in fact does not contain a Reebless foliation, or a transversely

oriented essential lamination. (We refer the reader to their article for definitions of these

terms, as we shall not make further use of them here). Furthermore, it was shown by

Fenley in [Fen02] that the manifolds RSSm(p/q) form ≤ −4 and |p−2q| = 1 do not admit

8a simply-connected, second countable but not necessarily Hausdorff 1-manifold

9We remark that this also implies that the fundamental group is not left-orderable, i.e. there exists

no total order of its elements which is invariant under left-multiplication (see for example [BRW05]).

5

any essential laminations, answering in the negative the long-standing question whether

all closed hyperbolic 3-manifolds admit essential laminations10. Essential laminations

can be regarded as a generalization of essential surfaces, and the existence (or non-

existence) of such objects has wide-reaching consequences for the topology of a manifold.

This raised the natural question whether these manifolds would at least virtually admit

an essential lamination, i.e. whether there is a finite cover with this property. Our

contribution to this question is to show that indeed many of the manifolds RSSm(p/q)

have virtually11 positive first Betti number12 , and as a consequence virtually possess

essential surfaces. We achieve this by applying a result of Masters ([Mas00]) on the

virtual homology of once-punctured torus bundles.

The organization of this thesis is as follows:

Chapter 1: This chapter contains various background material, terminology and nota-

tion needed throughout this thesis. In particular, short summaries of the following

topics are given: SL2(C) (and to some degree PSL2(C)) representation, charac-

ter and eigenvalue varieties; SL2(C) Culler-Shalen seminorms; some background

on basic algebraic geometry and elimination theory; group cohomology and its

relationship to the smoothness of the representation and character varieties. Fur-

thermore, some basic calculations of the fundamental group and the relevant affine

varieties for the (unfilled) Whitehead link exterior are given.

Chapter 2: In this chapter, we determine the set of all p-reps of the manifold Wp/q.

This is done by identifying the set of conjugacy classes of p-reps with a subset of the

subvariety C(Wp/q) of the representation variety, which then in turn is identified

with a certain subset of the eigenvalue variety of W . Solving the polynomial

10We remark that this was conjectured for m odd in [RSS03].

11A manifold is said to virtually satisfy a property, if it has a finite cover with that property.

Analogously, properties of its fundamental group are said to hold virtually, if the group has a finite-index

subgroup having these properties.

12The rank of its first homology with integer coefficients, b1(M) = rankH1(M ; Z)

6

system describing this subset using elimination theory leads to the definition of

the (Laurent) polynomial resp,q, whose roots then are shown to characterize the set

of p-reps. A detailed analysis of the properties and symmetries of these recursively

defined polynomials is given. In particular, it is conjectured (and proven for q = 1)

that all roots of resp,q (apart from 0 and ±1) are simple roots. Without assuming

this conjecture, we thus obtain an upper bound for the number of conjugacy classes

of p-reps from the degree of resp,q — while the truth of the conjecture would imply

that this bound is in fact attained.

Chapter 3: The minimal nonzero value of the Culler-Shalen seminorm of Wp/q is in-

vestigated and shown to be equal to the number of conjugacy classes of p-reps:

Using group cohomology calculations, it is shown that each character of a p-rep is

a smooth point on a non-trivial curve in the character variety of Wp/q, and that

(away from a certain finite collection of cases) it furthermore is a simple root of

the trace function fµ1 . This gives an upper bound for the minimal Culler-Shalen

seminorm in terms of the results of Chapter 2.

Chapter 4: The total Culler-Shalen seminorm is determined for p odd: First we de-

termine the seminorm explicitly for certain slopes which are known to give Seifert

fillings of Wp/q, counting their non-abelian SL2(C) characters. Using the upper

bound for the minimal seminorm obtained in Chapter 3, we then solve certain

linear systems which relate the seminorms obtained for the Seifert filling with the

coefficients determining the general form. In the listed cases, this in fact shows

that the upper bound for the minimal seminorm is attained, in turn showing that

the conjecture in Chapter 2 about the simplicity of roots is true when p is odd.

Chapter 5: Integer fillings of the Whitehead link exterior are examined. These are

once-punctured torus bundles with a certain type of monodromy. Some general

background on punctured torus bundles, their monodromies and their framings

is discussed. We then study closed manifolds obtained by further Dehn filling,

describing the results of [RSS03] and [Fen02]. We finish by applying the results

7

of [Mas00] to our situation, showing the virtual Z-representability of most of the

considered manifolds.

Chapter 1

Preliminaries

In this chapter, we summarize some background material which will be needed

in later chapters. The reader may therefore choose to skip through this chapter and

directly proceed to chapter 2, referring back only as needed. Most proofs of results in

this chapter will be omitted and can be found in the literature.

1.1 Some algebraic varieties associated to 3-manifolds

1.1.1 The SL2(C) and PSL2(C) representation varieties

We summarize some background material on the theory of representations of

3-manifold groups into SL2(C) and PSL2(C). An excellent introduction is [Sha02].

Let G be a Lie group and g its Lie algebra, and let Γ be a finitely presented1 group,

i.e. Γ = 〈 s1, . . . , sn | r1, . . . , rm 〉. Denote the space of all representations ρ : Γ → G by

R(Γ;G) := Hom(Γ;G). Note that this space can be identified with

R(Γ;G) = {(g1, . . . , gn) ∈ Gn | rj(g1, . . . , gn) = I (j = 1, . . . ,m)}, (1.1)

where I is the unit element in G. Hence we have a regular map r := (r1, . . . , rm) : Gn →

Gm and R(Γ;G) can be identified with r−1(I, . . . , I). The representation ρ is identified

with (g1, . . . , gn) iff ρ(si) = gi.

1We remark that the following arguments apply equally well for a finitely generated group, but

we will not need this generality here.

9

We will mostly be interested in the cases when G = SL2(C) or G = PSL2(C) and

Γ = π1(M) is the fundamental group of a 3-manifold. We set therefore

R(M) := R(π1(M); SL2(C)) (1.2)

and

R(M) := R(π1(M); PSL2(C)). (1.3)

Note that these spaces inherit an algebraic structure from SL2(C) (respectively PSL2(C))

as affine algebraic sets, and the induced structure does not depend on the group pre-

sentation. These spaces are called the SL2(C) and PSL2(C) representation varieties of

Γ. The space R(M) (resp. R(M)) carries two topologies, the Zariski and the complex

(or classical) topology (cf. [Sha94, Ch. II, §2.3]).

Two representations are equivalent if they differ by an inner automorphism of

SL2(C) (resp. PSL2(C)). A representation is irreducible if the only subspaces of C2

invariant under its image are trivial, i.e. the representation cannot be conjugated to a

representation by upper (or lower) triangular matrices. Otherwise a representation is

reducible. The reducible representations form a closed subset Rred(M) of R(M) (resp.

Rred(M) ⊂ R(M)). Denote the Zariski closure of its complement by Rirr(M) (resp.

Rirr(M)), i.e. Rirr(M) = R(M)−Rred(M), it contains all irreducible representations,

but in general also reducible representations. A representation ρ ∈ R(M) is strictly

irreducible it its image does not conjugate into N := {(a 00 a−1

),

(0 b

−b−1 0

)| a, b ∈ C∗}.

We will call a representation abelian, non-abelian, dihedral, etc. if its image has this

property.

In the following, we will specialize to the case when Γ = π1(M) is a group with

two generators and presentation 〈x, y | r1, . . . , rn 〉. We will now proceed to define two

subvarieties of R(Γ) which will prove to be very useful in applications.

10

1.1.1.1 A triangular subvariety

Define C(Γ) ⊂ R(Γ) to be the subvariety defined by the three equations ρ(x)2,1 =

0, ρ(y)1,2 = 0, ρ(y)2,1 = 1 fixing these three entries of the representation images. This

means that for ρ ∈ C(Γ),

ρ(x) =

s c

0 s−1

and ρ(y) =

u 0

1 u−1

. (1.4)

Now note that every irreducible2 representation ρ ∈ R(Γ) can be conjugated into

C(Γ). In fact, there are generically four choices for a base (b1, b2) of C2 with respect

to which ρ has the form (1.4). These can be obtained by first choosing a vector b′1

invariant under ρ(x), a vector b′2 invariant under ρ(y), and finally conjugating by a ma-

trix which is diagonal with respect to (b′1, b′2). The involutions (s, u, c) 7→ (s−1, u−1, c)

and (s, u, c) 7→(s, u−1, c+ (s− s−1)(u− u−1)

)generate a Kleinian four-group acting

on the set of possible bases for the normal form (1.4). It follows that C(Wp/q) projects

generically 4-to-1 to X(Wp/q).

1.1.1.2 A partially diagonal subvariety

In applications (e.g. the cohomology calculations in section 3.1.3), it will often be

useful to diagonalize the first generator image. Therefore define the subvariety D(Γ) ⊂

R(Γ) by the three equations ρ(x)1,2 = 0, ρ(x)2,1 = 0 and ρ(y)2,1 = 1. This means that

for ρ ∈ D(Γ),

ρ(x) =

s 0

0 s−1

and ρ(y) =

a ab− 1

1 b

. (1.5)

Note that only representations ρ ∈ R(Γ) which send the generator x to a non-parabolic

element conjugate into ρ ∈ D(Γ). The reducible representations in D(Γ) correspond to

the case ab = 1.

2On the other hand, C(Γ) also contains reducible representations (in the case that c = 0). This

includes certain non-abelian reducible representations, as discussed in section 1.1.1.3.

11

1.1.1.3 A note on non-abelian reducible representations

Let ρ ∈ R(Γ) be a non-abelian reducible representation. Up to conjugation we

can bring ρ into (lower) triangular form, and after adjusting with a diagonal matrix we

can assume that

ρ(x) =

s 0

c s−1

and ρ(y) =

u 0

d u−1

. (1.6)

We now have two cases:

1. If ρ(x) is not parabolic, i.e. s2 6= 1, we can diagonalize ρ(x) and hence assume

c = 0. This means that ρ can be conjugated into C(Γ) ∩D(Γ) in this case3.

2. If on the other hand ρ(x) is parabolic (s2 = 1), we must have u2 6= 1 and c 6= 0,

as otherwise ρ would be abelian. In particular, ρ cannot be conjugated into either

of C(Γ) or D(Γ).

1.1.2 The SL2(C) and PSL2(C) character varieties

The character of a representation ρ ∈ R(M) is the function χρ : π1(M) → C de-

fined by χρ(γ) := trace ρ(γ) for γ ∈ π1(M). Irreducible representations are determined

by characters up to equivalence.

The collection of characters of representations in R(M) can be regarded as an

affine algebraic set, which is called the SL2(C) character variety , denoted by X(M) (cf.

[CS83]). There is a regular map t : R(M) → X(M) taking representations to characters.

The analogous set for PSL2(C) characters, the PSL2(C) character variety X(M)

can be defined as the algebro-geometric quotient of R(M) under the natural action of

PSL2(C) by conjugation (cf. [BZ98]). We will denote all relevant objects for PSL2(C)

by placing a bar over the corresponding SL2(C) notation. As with the SL2(C) character

variety, there is a surjection t : R(M) → X(M) which is constant on conjugacy classes.

3We have d 6= 0, since ρ is non-abelian.

12

Furthermore, if ρ is an irreducible representation, then t−1 (t(ρ)

)is the orbit O(ρ) of ρ

under conjugation. The natural map q : X(M) → X(M) is finite-to-one, but in general

not onto.

We refer the reader to [BZ98] for more details.

1.1.3 The SL2(C) eigenvalue variety

The eigenvalue variety of a manifold whose boundary consists of a finite number

of tori was introduced by Tillmann in [Til05] as a generalization of the A-polynomial4

as follows.

Let M be a 3-manifold whose boundary consists of h disjoint tori T1, . . . , Th.

Given a presentation of π1(M) with n generators, introduce four affine coordinates

(representing the matrix entries in SL2(C)) for each generator, denoted by gij for i =

1, . . . , n and j ∈ {1, 2, 3, 4}. View R(M) as an affine algebraic set in C4n defined by an

ideal J in C[g11, . . . , gn4]. There are elements Iγ ∈ C[g11, . . . , gn4] for each γ ∈ π1(M)

such that Iγ(ρ) = trace ρ(γ) for each ρ ∈ R(M).

Choose a basis (µi, λi) for im (π1(Ti) → π1(M)). We will call µi the meridian and

λi the longitude of Ti. As µi and λi are words in the generators of π1(M), we can define

the following polynomial equations in the ring C[g11, . . . , gn4,m±11 , l±1

1 , . . . ,m±1h , l±1

h ]:

Iµi = mi +m−1i , (1.7a)

Iλi= li + l−1

i , (1.7b)

Iµiλi= mili +m−1

i l−1i (1.7c)

for i = 1, . . . , h. Let RE(M) be the variety in C4n × (C∗)2h defined by J together with

the above equations. Note that for each ρ ∈ R(M), the equations (1.7) have a solution

since commuting elements of SL2(C) always have a common invariant subspace. The

4Which is defined for manifolds with boundary consisting of a single torus.

13

natural projection p1 : RE(M) → R(M) is therefore surjective, and p1 is a dominating5

map.

The eigenvalue variety E(M) is the closure of the image of RE(M) under projec-

tion onto the coordinates (m1, l1, . . . ,mh, lh). It is therefore defined by an ideal of the

ring C[m±11 , l±1

1 , . . . ,m±1h , l±1

h ] in (C∗)2h.

An analogous variety can be defined for PSL2(C) representations, denoted by

E(M). Since we will not need it for the purposes of this thesis, we refer the interested

reader to [Til05] for more details.

1.2 Culler-Shalen seminorms

In this section, we briefly recall the construction of the Culler-Shalen norm devel-

oped in [CGLS87], and of its generalization to a seminorm by Boyer and Zhang in [BZ98].

We follow the exposition in [BZ98], restricting our attention to SL2(C) representations.

Let M be a connected, orientable, compact, irreducible and boundary-irreducible

3-manifold with boundary ∂M consisting of one torus. Unless noted otherwise, we will

generally suppress the base point of fundamental groups, π1(∂M) will be identified with

its image under the natural inclusion π1(∂M) → π1(M) (defined up to conjugation).

The first homology of the boundary with integer coefficients H1(∂M ; Z) will be denoted

by L and regarded as a lattice in the 2-dimensional real vector space V = H1(∂M ; R).

Using the Hurewicz homomorphism, we will often identify L with π1(∂M).

A slope r is the isotopy class of an essential, unoriented, simple closed curve on

∂M . It can be identified with a pair of elements ±δ ∈ L. Let α(r) be either one of ±δ.

We will often slightly abuse notation and simply write δ instead of α(r), e.g. writing

M(δ) instead of M(r) to denote the manifold obtained by Dehn filling M along the slope

r. A slope is called a boundary slope if it represents the isotopy class of the boundary

5A polynomial map φ : V → W between affine varieties is dominating if the smallest variety of

W containing φ(V ) is W itself. Thus, φ is dominating if its image is Zariski dense in W .

14

components6 of an essential surface S ⊂ M . A boundary slope is called strict if it is

the boundary slope of an essential surface S which is neither a fibre or a semi-fibre in a

fibre or semi-fibre bundle structure on M .

For two elements δ1, δ2 ∈ L, we denote their geometric intersection number, called

their distance, by ∆(δ1, δ2).

Let X0 ⊂ X be a curve in the character variety X = X(M). Call a curve of the

character variety non-trivial if it contains the character of an irreducible representation.

For γ ∈ π1(M), define the regular function Iγ : X0 → C by Iγ(χρ) = χρ(γ) =

trace ρ(γ). Use this to define the map fγ := I2γ − 4, which is also regular and can hence

be pulled back to the smooth projective model7 X0 of X0 (we will denote this map also

by fγ).

For γ ∈ L, define ‖γ‖X0as the degree of the map fγ : X0 → CP1. This can be

extended naturally to define a function

‖·‖X0: H1(∂M ; R) → [0;∞) (1.8)

on V := H1(∂M ; R).

This function is a seminorm on H1(∂M ; R), called the Culler-Shalen seminorm

associated to the curve X0 (cf. [CGLS87, Proposition 1.1.2] for the case of a norm, and

[BZ98, Section 5] for the general case). Recall that a seminorm on a 2-dimensional real

vector space is either a norm, a nonzero indefinite seminorm, or identically zero.

If fγ is non-constant on X0 for some γ ∈ L, define

sX0 := min{‖γ‖X0| γ ∈ L, γ 6= 0, fγ not constant on X0}, (1.9)

otherwise set sX0 := 0. We call sX0 the minimal seminorm on X0.

6On one component of ∂M .

7The unique non-singular projective variety birationally equivalent to the curve X0, cf. [Sha94,

Ch. II, §4.5 and 5.3].

15

Define the fundamental ball of ‖·‖X0to be

BX0 = {v ∈ V | ‖v‖X0≤ sX0}. (1.10)

The basic properties of ‖·‖X0were developed by Culler and Shalen, especially in

the case where it is a norm (cf. [CGLS87]). The following three propositions summarize

some of these properties when ‖·‖X0is a norm, nonzero indefinite seminorm or identically

zero.

Proposition 1.2.1 ([BZ98, Proposition 5.2])

If ‖·‖X0is a norm on H1(∂M ; R), then the following hold:

1. The curve X0 is non-trivial, i.e. it contains the character of an irreducible repre-

sentation.

2. Each of the functions fγ for 0 6= γ ∈ L is nonconstant on X0.

3. The fundamental ball BX0 is a compact, convex, finite-sided polygon which is bal-

anced, i.e. BX0 = −BX0.

4. Each vertex of BX0 is a rational multiple of a primitive element of L which corre-

sponds to a boundary slope on ∂M . �

An important example is when the interior of M admits a complete hyperbolic metric of

finite volume. If X0 is any irreducible component of X(M) which contains the character

of a discrete, faithful representation, then X0 has dimension 1 (cf. [CS84]) and ‖·‖X0is

a norm (cf. [CGLS87]).

Proposition 1.2.2 ([BZ98, Proposition 5.4])

If ‖·‖X0is a nonzero indefinite seminorm on H1(∂M ; R), then the following hold:

1. There is a unique slope r for which fr|X0 is constant.

2. If γ ∈ L, then the function fγ is constant on X0 iff γ = nα(r) for some n ∈ Z.

16

3. The fundamental ball BX0 is an infinite band centred on the line through 0 and

α(r). In particular, ‖γ‖X0= ∆(r′, r)sX0 for any γ = α(r′) ∈ L.

4. The slope r is a boundary slope. �

The curve X0 is called an r-curve in the situation of Proposition 1.2.2.

Proposition 1.2.3 ([BZ98, Proposition 5.5]) If ‖·‖X0is identically zero, then each of

the functions fγ for γ ∈ L is constant on X0. Hence there is an essential, closed surface

in M . �

This possibility can therefore not arise when the manifold M is small, i.e. when it does

not contain any closed essential surfaces. In that case, we have that dim X(M) ≤ 1 (cf.

[CCG+94]) and it is natural to consider the sum of the Culler-Shalen seminorms over

all non-trivial curves8 Xi ⊂ X(M), which we will call the total Culler-Shalen seminorm

:

‖γ‖ :=∑i

‖γ‖i (1.11)

The power of the machinery of Culler-Shalen seminorms is maybe best illustrated

in the following results on cyclic and finite Dehn fillings. Let X0 be a curve in X.

Theorem 1.2.4 ([CGLS87, Corollary 1.1.4])

If α is not a boundary slope and π1(M(α)) is cyclic, then ‖α‖X0= sX0. �

Theorem 1.2.5 ([BZ96, Theorem 2.3]) If α is not a boundary slope and π1(M(α)) is

finite, then ‖α‖X0≤ max(2sX0 , sX0 + 8). �

Denote by Πx(fβ) the order of the pole of fβ at x. Call a boundary slope β

and an ideal point x of X associated if β is the unique slope such that Πx(fβ) = 0,

i.e. such that fβ does not have a pole at x. Note that it is possible for a slope to be

associated to several ideal points, and it is also possible for an ideal point x to have no

8One-dimensional irreducible components

17

associated slopes, i.e. Πx(fδ) = 0 for each δ ∈ L. A boundary slope which is associated

to an ideal point is called strongly detected by the character variety. It can be verified

from Proposition 1.2.7 of [CGLS87] (cf. also Proposition 4.4 of [BZ98]) that a strongly

detected slope is strict if the irreducible component of X containing x contains a strictly

irreducible character.

Lemma 1.2.6 ([BZ96, Lemma 6.1]) The pairs of vertices of the fundamental polygon

B of the Culler-Shalen seminorm correspond precisely to the distinct strongly detected

boundary slopes. �

The following lemma will be of great use for us in chapter 4, as it enables us to

determine the Culler-Shalen seminorm from only a few known values:

Lemma 1.2.7 ([BZ96, Lemma 6.2]) If βj are the boundary slopes of M and Xi ⊂

X(M) is o a non-trivial curve, then

‖γ‖Xi=

∑j

aij∆(γ, βj),

where the aij are non-negative, even integers independent of γ ∈ π1(M). �

For the total seminorm, we therefore get

‖γ‖ =∑j

aj∆(γ, βj), (1.12)

with non-negative even integers aj . We will determine these integers for the manifolds

Wp/q in chapter 4.

We will often make implicit use of the following consequence of the previous

lemma:

Lemma 1.2.8 ([BZ96, Lemma 6.3])

1. For any 0 6= δ ∈ L, ‖δ‖ is a positive even integer.

2. Suppose that the fundamental polygon B has k vertices. Then s ≥ 2k − 2, and if

∂B contains a lattice point which is not a vertex of B, then s ≥ 2k. �

18

1.3 Elimination theory

We summarize some basic facts from algebraic geometry, focusing on elimination

theory. A good reference is [CLO97]. We will restrict attention to the polynomial ring

C[x1, . . . , xn] over the complex numbers.

1.3.1 Monomial orders and Groebner bases

Let I be an ideal in the polynomial ring C[x1, . . . , xn]. A monomial order on

C[x1, . . . , xn] is any total ordering > on Zn+, or equivalently on the set of monomials

xα, α ∈ Zn+, which is a well-ordering9 and which satisfies α > β ⇒ α + γ > β + γ for

each γ ∈ Zn+. A commonly used monomial order is the (pure) lexicographic order, or lex

order for short, where α > β if in the vector difference α−β ∈ Zn, the left-most nonzero

entry is positive. Let f =∑

α aαxα ∈ C[x1, . . . , xn]. With respect to a fixed monomial

order, define the multidegree of f to be multideg(f) = max{α ∈ Zn+ | aα 6= 0}. The

leading term of f is then LT(f) = amultideg(f)xmultideg(f). Define LT(I) as the set of all

leading terms of all polynomials in the ideal I.

A finite subset G = {g1, . . . , gs} of an ideal I is said to be a Groebner basis

if 〈LT(g1), . . . ,LT(gs) 〉 = 〈LT(I) 〉. By the Hilbert basis theorem, there is a finite

generating set for each ideal I, i.e. I = 〈g1, . . . , gs〉 for some g1, . . . , gs ∈ I. Furthermore,

these generators can be chosen to form a Groebner basis.

Recall that there is a general division algorithm in C[x1, . . . , xn]: Dividing f ∈

C[x1, . . . , xn] by f1, · · · , fs ∈ C[x1, . . . , xn] means expressing f as f = a1f1+· · ·+asfs+r,

where the “quotients”a1, . . . , as and r lie in C[x1, . . . , xn]. The result depends in general

on the chosen monomial order and on the way f1, . . . , fs are listed.

This ambiguity disappears when dividing by the elements of a Groebner basis:

Proposition 1.3.1 (cf. [CLO97, Ch. 2, §6, Proposition 1])

Let G = {g1, . . . , gs} be a Groebner basis for an ideal I ⊂ C[x1, . . . , xn] and let f ∈

9Every nonempty subset has a smallest element.

19

C[x1, . . . , xn]. Then there is a unique r ∈ C[x1, . . . , xn] such that

1. No term of r is divisible by any of LT(g1), . . . ,LT(gs).

2. There is g ∈ I such that f = g + r.

In particular, r is the remainder on division of f by G no matter how the elements of

G are listed when using the division algorithm. �

The remainder r is sometimes called the normal form of f .

1.3.2 The Elimination and Extension Theorems

Given I = 〈 f1, . . . , fs 〉 ⊂ C[x1, . . . , xn], define the k-th elimination ideal Ik ⊂

C[xk+1, . . . , xn] to be Ik := I ∩C[xk+1, . . . , xn]. Thus, Ik consists of all consequences of

f1 = · · · = fs = 0 which eliminate the variables x1, . . . , xk.

Theorem 1.3.2 (cf. [CLO97, Ch. 3, §1, Theorem 2]) Let I ⊂ C[x1, . . . , xn] and let G

be a Groebner basis of I with respect to lex order where x1 > x2 > · · · > xn. Then,

for every 0 ≤ k ≤ n, the set Gk := G ∩ C[xk+1, . . . , xn] is a Groebner basis of the k-th

elimination ideal Ik. �

Consider the affine variety V (I) defined by I,

V (I) = {(a1, . . . , an) ∈ Cn | f(a1, . . . , an) = 0 ∀f ∈ I }. (1.13)

The basic idea to describe points of V (I) is to build up solutions one coordinate at a

time. Fixing some k between 1 and n, we obtain the elimination ideal Ik and associated

solutions (ak+1, . . . , an) ∈ V (Ik), which we call partial solutions of the original system

of equations. To extend this to a complete solution in V (I), one needs to find ak such

that (ak, ak+1, . . . , an) lies in the variety V (Ik−1) of the next elimination ideal. This is

not always possible, i.e. there are partial solutions which do not extend. Geometrically,

if πk : Cn → Cn−k denotes the projection map onto the last n− k coordinates, we have

that πk(V (I)) ⊂ V (Ik), and this might be a strict inclusion. Nevertheless, we have:

20

Theorem 1.3.3 (cf. [CLO97, Ch. 3, §1, Theorem 3]) Let I = 〈 f1, . . . , fs 〉 ⊂ C[x1, . . . , xn]

and let I1 be the first elimination ideal of I. For each 1 ≤ i ≤ s, write fi in the form

fi = gi(x2, . . . , xn)xNi1 + terms in which x1 has degree < Ni,

where Ni ≥ 0 and gi ∈ C[x2, . . . , xn] is nonzero. Suppose that we have a partial solution

(a2, . . . , an) ∈ V (I1). If (a2, . . . , an) 6∈ V (g1, . . . , gs), then there exists a1 ∈ C such that

(a1, a2, . . . , an) ∈ V (I). �

1.3.3 Resultants

A special case is when I ⊂ C[x1, . . . , xn] is generated by only two polynomials f

and g. Suppose f and g are of positive degree in x1, and write them in the form

f = a0xl1 + · · ·+ al, a0 6= 0, g = b0x

m1 + · · ·+ bm, b0 6= 0, (1.14)

where ai, bi ∈ C[x2, . . . , xn]. Define the resultant of f and g with respect to x1 to be

the determinant of the so-called Sylvester matrix

Res(f, g, x1) := det

a0 b0

a1 a0 b1 b0

a1

. . . b1. . .

.... . . a0

.... . . b0

... a1

... b1

al bm

al

... bm...

. . .. . .

al bm

(m+l)×(m+l)

. (1.15)

We then have that Res(f, g, x1) is in the first elimination ideal 〈 f, g 〉∩C[x2, . . . , xn]. Fur-

thermore, Res(f, g, x1) = 0 if and only if f and g have a common factor in C[x1, . . . , xn]

which has positive degree in x1. In other words, if (p1, . . . , pm) is a common root of f

and g, then the partial solution (p2, . . . , pm) is a root of Res(f, g, x1). Inversely, a given

root P = (p2, . . . , pm) of Res(f, g, x1) can be extended to a solution (p1, p2, . . . , pm) of

f = g = 0 under the condition that a0 and b0 do not vanish at P .

21

This fact can iteratively be applied to a collection of polynomials, and to eliminate

more than one variable.

1.4 Group cohomology

We collect some basic facts about group cohomology. Good references are [Bro82]

and [Wei64], we also recommend the articles [HPSP01] and [HP04] for a good summary

in our context.

First recall that the first cohomology group of a 3-manifold can be identified with

the first cohomology of its fundamental group:

Lemma 1.4.1 (cf. [HP04, Lemma 3.1]) Let A be a π1(M)-module and let X be any

CW-complex with π1(X) ∼= π1(M). There are natural morphisms Hi(X;A) → Hi(π1(M);A)

that are isomorphisms for i = 0, 1 and a surjection for i = 2. In cohomology there are

natural morphisms H i(π1(M);A) → H i(X;A) that are isomorphisms for i = 0, 1 and

an injection for i = 2. �

In the following let Γ = π1(M) be the fundamental group of a 3-manifold M and

assume that it is finitely presented.

Let G be a Lie group and g its Lie algebra, and denote the adjoint homomorphism

by Ad: G→ Aut(g). Let ρ ∈ R(Γ;G) be a representation. The Lie algebra g of G can

then be viewed as a (left) Γ module via the map Ad ◦ρ, i.e. for γ ∈ Γ and g ∈ G we

have the left action γ · g := Adρ(γ)(g).

In our situation, G = SL2(C), g = sl2(C) (the set of complex 2× 2 matrices with

trace zero), and the adjoint homomorphism acts by conjugation, i.e. for A ∈ SL2(C)

and U ∈ sl2(C) we have Ad(A)(U) = AUA−1.

We will write slρ2 to denote sl2(C) viewed as a Γ module via the Ad ◦ρ left-

action. Denote by C∗(Γ; slρ2) the space of cochains with coefficients in slρ2. Similarly, let

B∗(Γ; slρ2) and Z∗(Γ; slρ2) the coboundaries and cocycles, and denote by H∗(Γ; slρ2) the

cohomology group with coefficients in slρ2.

22

Note that a cocycle d ∈ Z1(Γ; slρ2) is a map d : Γ → slρ2 satisfying

d(γ1γ2) = d(γ1) + γ1 · d(γ2) for all γ1, γ2 ∈ Γ, (1.16)

using the short notation for the Ad ◦ρ action of Γ on sl2(C). A cochain b : Γ → slρ2 is a

coboundary if there exists an U ∈ sl2(C) such that b(γ) = γ · U − U for all γ ∈ Γ.

It was an observation of Weil ([Wei64], cf. also [Por97]) that the Zariski tangent

space TZarρ (R(Γ)) at a point ρ ∈ R(Γ) can be identified with a subspace of the space of

1-cocycles Z1(G; slρ2), and thus dimTZarρ (R(Γ)) ≤ dimZ1(Γ; slρ2).

Denote the local dimension of R(Γ) at ρ by dimρ R(Γ), it is the maximum of the

dimensions of all algebraic components of R(Γ) containing ρ (cf. [Sha94, Ch. II]).

Lemma 1.4.2 Let ρ ∈ R(Γ). If dimρ R(Γ) = dimZ1(Γ; slρ2), then ρ is a smooth point

of the representation variety R(Γ) and ρ is contained in a unique component of R(Γ)

of dimension dimZ1(Γ; slρ2).

Proof. For every ρ ∈ R(Γ), we have

dimρ R(Γ) ≤ dimTZarρ (R(Γ)) ≤ dimZ1(Γ; slρ2).

Thus dimρ R(Γ) = dimZ1(Γ; slρ2) implies dimρ R(Γ) = dimTZarρ (R(Γ)), i.e. ρ is a simple

point of R(Γ) (cf. [Sha94, Ch. II, §1.4]). The conclusion now follows from Theorem 6 of

[Sha94, Ch. II, §2.2]. �

Important for us will be the following consequence for the character variety: Let

Rirr(Γ) ⊂ R(Γ) be the set of irreducible representations and Xirr(Γ) ⊂ X(Γ) the set

of characters of such representations. Proposition 3.2 of [Por97] shows that the map

t : Rirr(Γ) → Xirr(Γ) is an analytic bundle whose fibres are the orbits (under conjuga-

tion) O(ρ) where ρ ∈ Rirr(Γ). It then follows (cf. Proposition 3.5 of [Por97]) that the

Zariski tangent space TZarχρ

(X(Γ)) of the character variety at the character χρ of an irre-

ducible representation ρ ∈ Rirr(Γ) can be identified with a subspace of the cohomology

23

group H1(Γ; slρ2). Furthermore, this identification is such that the following diagram

commutes:

TZarρ O(ρ) � � //

∼=��

TZarρ R(Γ) // //

��

TZarχρ

X(Γ)

��

B1(Γ; slρ2)� � // Z1(Γ; slρ2) // // H1(Γ; slρ2).

(1.17)

We therefore have

dimχρ X(Γ) ≤ dimTZarχρ

(X(Γ)) ≤ dimH1(Γ; slρ2), (1.18)

which immediately implies the following lemma:

Lemma 1.4.3 Let χρ ∈ Xirr(Γ). If dimχρ X(Γ) = dimH1(Γ; slρ2), then χρ is a smooth

point of the character variety X(Γ) and χρ is contained in a unique component of X(Γ)

of dimension dimH1(Γ; slρ2).

Proof. Identical to that of Lemma 1.4.2, using the inequalities (1.18). �

When ρ is a non-abelian reducible representation, the situation is more subtle

and was studied in [Boy02b]:

Theorem 1.4.4 (Theorem A of [Boy02b]) Suppose that ρ ∈ R(Γ) is a non-abelian

representation lying in Rirr(Γ) for which Z1(Γ; slρ2) ∼= C4. Then

1. ρ is a simple point of R(Γ) and the algebraic component of R(Γ) which contains

it is 4-dimensional;

2. χρ is a simple point of Xirr(Γ) and the algebraic component of Xirr(Γ) which con-

tains it is a curve;

3. there is an analytic 2-disk D, smoothly embedded in R(Γ) and containing ρ, such

that the projection t|D is an analytic isomorphism onto a neighborhood of χρ in

Xirr(Γ); �

The following result of W. Thurston is useful in determining a lower bound for

dim X(M) of a manifold M with boundary:

24

Proposition 1.4.5 (W. Thurston, cf. Prop. 3.2.1 of [CS83])

Let M be a compact orientable 3-manifold, and let h be the number of torus components

of its boundary ∂M . Let ρ0 ∈ R(M) be an irreducible representation such that for each

torus component T of the boundary, ρ0 (im(π1(T ) → π1(M))) 6⊂ {±I}. Let R0 ⊂ R(M)

be an irreducible component containing ρ0. Then its image X0 := t(R0) under the

projection t : R(M) → X(M) has dimension dim X0 ≥ h− 3χ(M). �

Note that if ∂M consists only of tori, we always have χ(M) = 0 and so we simply get

dim X0(M) ≥ h. (1.19)

1.5 The Whitehead link exterior

In this section, we summarize some well-known results about the Whitehead link

exterior.

1.5.1 Fundamental group

a

b

dec

Figure 1.1 The right-handed Whitehead link

Reading off a Wirtinger presentation from the diagram in figure 1.1, we get the

following presentation of the fundamental group of the complement of the right-handed

Whitehead link:

π1(Wrh) = 〈 a, b, c, d, e | da = bd, bc = db, be = ea, ea = ac 〉 (1.20)

The preferred longitudes (cf. [Rol76] or [BZ03]) can be read off this diagram as λ0 =

d−1e and λ1 = b−1cac−1. By applying Tietze transformations, we can simplify this

25

presentation to obtain

π1(Wrh) = 〈 a, b, c, d, e | da = bd, bc = db, be = ea, ea = ac 〉

=⟨a, c | aca−1c−1a−1caca−1c−1acac−1a−1c−1 = 1

⟩=

⟨µ0, µ1 | µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0µ1 = µ1µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0

⟩ (1.21)

where a = µ0 and c = µ1. In these generators, the longitudes become

λ0 = µ0µ1µ0µ−11 µ−1

0 µ−11 µ0µ1µ

−20 = µ1µ0µ

−11 µ−1

0 µ−11 µ0µ1µ

−10 (1.22)

(since it commutes with µ0) and

λ1 = µ0µ1µ−10 µ−1

1 µ−10 µ1µ0µ

−11 . (1.23)

Note that the relation can also be written as µ0λ0 = λ0µ0 or µ1λ1 = λ1µ1,

corresponding to the fact that the meridian and longitude of each boundary component

commute.

1.5.2 Subvarieties of the representation variety

1.5.2.1 The triangular subvariety

We will now determine the subvariety C(Wrh) ⊂ R(Wrh) (cf. section 1.1.1.1).

Using equation (1.4) with x = µ0 and y = µ1, a straightforward calculation shows that

the relation w1 := µ0µ1µ−10 µ−1

1 µ−10 µ1µ0µ1 = µ1µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0 =: w2 translates

into the condition

0 = ρ(w1)− ρ(w2) =

− cu2s2

f c(u2−1)u3s2

f

s2−1u2s3

f cu2s2

f

, (1.24)

where

f(s, u, c) = (s− s−1)(u− u−1) + c(s−2u−2 − u−2 − s−2 + 4− s2 − u2 + s2u2)

+c2(2s−1u−1 − su−1 − s−1u+ 2su) + c3.(1.25)

This shows that C(Wrh) ⊂ R(Wrh) is defined by the equation f(s, u, c) = 0. Note that

this computation can also be found in [Til02, section 5.1.9.].

26

1.5.2.2 The partially diagonal subvariety

We will now determine the defining equations of the subvariety D(Wrh) ⊂ R(Wrh)

(cf. section 1.1.1.2). For a given ρ ∈ D(Wrh), the group relation translates into the

matrix equation 0 s−4(s− 1)2(s+ 1)2r1

s−4(ab− 1)(s− 1)2(s+ 1)2r1 0

=

0 0

0 0

, (1.26)

where

r1 = s4a2b− s4a− a2bs2 + s2ab2 + b− ab2. (1.27)

The defining equation for D(Wrh) is therefore (s2 − 1)r1 = 0. If we exclude s = ±1

(which would give abelian representations), we find that the remaining representations

in D(Wrh) have to satisfy r1 = 0.

All polynomials in a, b and s relevant in our context can hence be regarded as

elements in the ring C[a, b]/ 〈 r1 〉 containing the coordinate ring of D. Unless otherwise

noted, we will always assume a pure lexicographic order with a > b > s and use this to

find the unique normal form f ′ with respect to the ideal 〈 r1 〉 for each polynomial f (cf.

section 1.3). We will write f ≡r1 f ′ in this case.

Note that since µ0 and λ0 commute and ρ(µ0) 6= ±I, the matrix ρ(λ0) also has

to be diagonal. In fact we find

ρ(λ0) =

s−4δ1 s−4(s− 1)(s+ 1)r1

s−2(s− 1)(s+ 1)(ab− 1)r1 s−2δ2

≡r1

s−4(a2s4 − a2s2 + abs2 − ab+ 1) 0

0 s−2(−s4b2 + s2b2 − s6ba+ s6 + s4ba)

=:

t 0

0 t′

(1.28)

Furthermore, note that tt′ ≡r1 1 as expected. In fact, we find that bt ≡r1 a, so that we

27

have

t ≡r1a

b. (1.29)

1.5.3 The eigenvalue variety

We will now determine the defining equations of the eigenvalue variety E(Wrh)

(cf. section 1.1.3), following [Til02, section 5.4.3].

Let ρ ∈ C(Wrh). For γ ∈ π1(Wrh), let

θγ : C(Wrh) → C

ρ 7→ ρ(γ)1,1(1.30)

be the map taking ρ to the upper left entry of the image ρ(γ). Now define the eigenvalue

map to be

e : C(Wrh) → E(Wrh)

ρ 7→ (θµ0(ρ), θλ0(ρ), θµ1(ρ), θλ1(ρ)) .(1.31)

Note that this is well-defined, as each ρ ∈ C(Wrh) is triangular on the peripheral sub-

group.

Note that by definition θµ0(ρ) = s and θµ1(ρ) = u. The eigenvalues of the

longitudes are given by the rational functions10

t = θλ0(ρ) = s−2 − s−2u2 + u2 + c(2s−1u+ s−3u−1 − s−3u

)+ c2s−2, (1.32a)

v = θλ1(ρ) = u−2 − u−2s2 + s2 + c(2u−1s+ u−3s−1 − u−3s

)+ c2u−2. (1.32b)

Let f1 = f(s, u, c) be the defining equation of C(Wrh) from equation (1.25), and

let f2 (resp. f3) be the equation obtained from (1.32a) by multiplying by s3u (resp.

from (1.32b) by multiplying by su3). Using elimination theory to eliminate the variable c

from these equations, we find that E0(Wrh) is defined by Res(f1, f2, c) = Res(f1, f3, c) =

10We give the reduced forms, obtained by expanding ρ(λi) (i = 0, 1) and replacing c3 us-

ing f(s, u, c) = 0 from equation (1.25). Furthermore, we use λ1 = µ0µ1µ−10 µ−1

1 µ−10 µ1µ0µ

−11 =

µ−11 µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0 (cf. equation (1.23)) to obtain the symmetrical form for v shown here.

28

Res(f2, f3, c) = 0. Simplifying and eliminating redundant factors gives us the three

defining equations of E0(Wrh) as

h1 = t− s2t+ s2t2 − s4t2 − u2 − 2s2tu2 + s4tu2 − t2u2 + 2s2t2u2

+ s4t3u2 + tu4 − s2tu4 + s2t2u4 − s4t2u4,

h2 = s2 − v − s4v + u2v + 2s2u2v + s4u2v − s2u4v + s2v2 − u2v2

− 2s2u2v2 − s4u2v2 + u4v2 + s4u4v2 − s2u4v3,

h3 = s4t− s6t− s2tu2 + s4tu2 + s6t2u2 − s2u2v

+ u4v + s2u4v − 2s4tu4v − u6v + s2u6v2.

(1.33)

Recall that two irreducible varieties V ⊂ Cm and W ⊂ Cn are said to be bira-

tionally equivalent , in symbols V 'W , if there exist rational mappings φ : V →W and

ψ : W → V such that φ ◦ ψ is equal to the identity map 1W where it is defined, and

similarly for ψ ◦ φ. The following result was shown in [Til02]:

Lemma 1.5.1 (Lemma 5.6 in [Til02]) The varieties C(Wrh) and E0(Wrh) are bira-

tionally equivalent. �

In particular it was shown that the map e : C(Wrh) → E0(Wrh) (cf. (1.31)) has degree

one, and determines an inverse mapping

e−1 : E0(Wrh) → C(Wrh),

(s, t, u, v) 7→(s, u,

s2(t− 1) + u2(1− v)su−1 − s−1u

).

(1.34)

This map is not defined when s2 = u2, but one can determine the following two inverse

maps which are defined on two open sets which cover all but eight points:

e−11 : E0(Wrh) → C(Wrh), (s, t, u, v) 7→

(s, u,

(s2 − v)(1− u2)su(1 + v)

), (1.35a)

e−12 : E0(Wrh) → C(Wrh), (s, t, u, v) 7→

(s, u,

(u2 − t)(1− s2)su(1 + t)

). (1.35b)

Four of the points where these two maps are not defined correspond to the discrete

faithful representations associated to the complete hyperbolic structure of Wrh, here

parametrized by t = v = −1 and s2 = u2 = 1. The remaining four points are given by

t = v = −1 and s2 = u2 = −1.

Chapter 2

The set of p-reps of a once-filled Whitehead link exterior

Throughout this chapter, let W := Wrh be the exterior of the right-handed White-

head link in S3, and let µ0, λ0 and µ1, λ1 be the meridians and longitudes of the two

boundary tori as in section 1.5.1. Let Wp/q := Wrh(p/q, · ) be the manifold obtained

from W by Dehn filling the first boundary torus along the slope p/q, i.e. by attaching

a solid torus by identifying its meridian with the curve µp0λq0. Here p and q are coprime

integers, and unless otherwise noted we will fix the convention q > 0.

The manifolds Wp/q constitute a rich class of 3-manifolds with torus boundary

which contains important infinite families of well-known manifolds such as the twist

knot exteriors (for p/q = 1/n, n ∈ Z) or an important class of punctured-torus bundles

(for p/q = m, m ∈ Z). Note that for example the figure-8 knot complement (p/q = −1)

and its so-called sister manifold (p/q = 5) are part of this family.

Recall from section 1.5.1 that the fundamental group of the Whitehead link ex-

terior admits the presentation

π1(W ) =⟨µ0, µ1 | µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0µ1 = µ1µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0

⟩(2.1)

which then gives the following presentation for the fundamental group of the filled

manifold:

π1(Wp/q) =⟨µ0, µ1 | µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0µ1 = µ1µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0,

µp0λq0 = 1

⟩.

(2.2)

30

Here µ0 and µ1 are the meridian of the first and second torus boundary, respectively.

The corresponding longitudes λ0 and λ1 are given in equations (1.22) and (1.23).

We are interested in representations of this group which send the meridian µ1 to

a parabolic1 element, and which have non-abelian image in SL2(C). Since µ1 and λ1

commute, this implies that ρ(λ1) is either ±I or also parabolic. Following Riley (cf.

[Ril72] and [Ril75]) we call such a representation a SL2(C) p-representation, or short

p-rep2.

These representations are interesting for many reasons. For us, their main impor-

tance will be that the count of their characters will enable us to determine the minimal

Culler-Shalen norm of Wp/q, which we will do in chapter 3. Another reason for their

importance lies in their connection with the hyperbolic structure on Wp/q, if it exists.

Recall that since W itself admits a complete hyperbolic metric of finite volume, the

same is true for all but a finite number of exceptional fillings Wp/q (cf. [Thu79]). The

holonomy representation ρ0 ∈ PSL2(C) associated to the hyperbolic structure then lifts

to give SL2(C) p-reps which are also discrete and faithful. Completely understanding

the set of all p-reps for a given hyperbolic filling of W therefore can be useful to identify

the discrete faithful representations associated to the hyperbolic structure3. We will

reserve this question for the future, concentrating on the application to Culler-Shalen

norms for the purposes of this thesis.

In [Ril72] and several later papers, Riley gives a general method to find all p-reps

of a knot or link group4. Exhibiting a set of certain normal forms for the images of

the generators he shows that the existence of a p-rep is then determined by a finite

number of polynomial equations (which are obtained by requiring the validity of the

1A matrix A 6= ±I with | traceA| = 2.

2Note that Riley used the name p-rep for PSL2(C) representations only, whereas a SL2(C)

representation which projects to a p-rep was called sl-rep, c.f. [Ril75]

3See for example the work of Hoste and Shanahan in [HS01], where they solve this problem for

all twist knots and proceed to compute the trace fields of these manifolds.

4His method applies in fact to any finitely presented group of deficiency 1.

31

group relations for the representation images). In the special case of a 2-bridge knot

(which admits a presentation with only two generators and one relation), he shows that

this system of polynomials consists in fact of only one polynomial (which he calls the

rep polynomial), and that all p-reps are therefore characterized up to conjugacy by the

roots of this polynomial.

In our situation, it is in general a difficult problem to simplify the group pre-

sentation (2.2) for a generic filling slope p/q to a presentation which lends itself to an

application of Riley’s method5. When this is indeed possible, the best one can in general

expect is to find a system of several polynomials characterizing the set of p-reps.

By taking a “detour” through the eigenvalue variety of the unfilled Whitehead

link exterior, we now find that the (irreducible) p-reps of the manifolds Wp/q can indeed

be characterized by the roots of a single one-variable polynomial, and that the set of

all p-reps can thus be determined in all generality. In some sense this can be regarded

as a generalization of Riley’s method, and it is the author’s belief that this method can

ultimately be extended to other once-filled 2-bridge link exteriors, and possibly also to

other 2-component link exteriors.

The organization of this chapter is as follows: In section 2.1, we will describe

how to identify the set of p-reps of the manifolds Wp/q with a certain subset of the

eigenvalue variety E(W ) of the unfilled Whitehead link. In section 2.2, we will find a

parametrization of this subset and describe its components corresponding to reducible

and irreducible representations. In section 2.3, we will then determine the solutions

(and their number) corresponding to reducible p-reps. In section 2.4, we will show

how the conjugacy classes of all irreducible p-reps can be characterized by the complex

roots of a certain one-variable polynomial resp,q which depends on the filling slope p/q.

In section 2.5, we will put together the results of the previous sections to obtain the

5One infinite family of fillings for which this is indeed possible is given by the twist knot fillings

(for p = 1). This is due to the fact that the twist knots themselves are 2-bridge knots. This has been

successfully used by Hoste and Shanahan in [HS01] to determine the rep polynomial.

32

count of conjugacy classes of p-reps. Finally, section 2.6 gives a detailed analysis of the

(Laurent) polynomials resp,q and of their roots.

The author would like to thank Stephan Tillmann for suggesting the “detour”

through the eigenvalue variety.

2.1 Detour through the eigenvalue variety

We begin by remarking that we can view all representations ρ ∈ R(Wp/q) as rep-

resentations of the fundamental group of the unfilled Whitehead link exterior: more

precisely, each representation ρ ∈ R(Wp/q) induces a representation ρ′ ∈ R(W ) sat-

isfying ρ′(µp0λq0) = I — and conversely, each such representation ρ′ factors through

π1(Wp/q). When the context is clear, we will sometimes (slightly) abuse notation and

denote both representations by ρ.

In particular, each (non-abelian) p-rep ρ ∈ R(Wp/q) induces a non-abelian repre-

sentation ρ′ ∈ R(W ) which still sends µ1 to a parabolic element. We will call such a

representation a partial p-rep, since it is only a p-rep restricted to one of the boundary

tori.

Furthermore, each p-rep ρ (respectively partial p-rep ρ′) can in fact be conjugated

into the subvariety C(Wp/q) (respectively C(W )), i.e. we can assume up to conjugation

that

ρ(µ0) =

s c

0 s−1

and ρ(µ1) =

u 0

1 u−1

(2.3)

for s, u ∈ C∗ and c ∈ C: This is clear when the representation is irreducible (cf.

section 1.1.1), so assume that ρ ∈ R(Wp/q) is a non-abelian reducible representation

which sends µ1 and λ1 to parabolic elements. Up to conjugation, we then have

ρ(µ0) =

s 0

c s−1

and ρ(µ1) =

u 0

1 u−1

, (2.4)

where u2 = 1. But then s2 6= 1, as otherwise ρ were abelian. We conclude that we

can in fact assume c = 0, and that ρ can hence be conjugated into C(Wp/q) (see also

section 1.1.1.1).

33

Denote by P ⊂ C(Wp/q) the subset of p-reps of π1(Wp/q). Note that by con-

struction, C(Wp/q) projects generically 4-to-1 to X(Wp/q), corresponding to the action

of the Kleinian four-group on the set of possible bases for the normal form (2.4) (see

section 1.1.1.1). This group is generated by the inversions (s, u, c) 7→ (s−1, u−1, c) and

(s, u, c) 7→(s, u−1, c+ (s− s−1)(u− u−1)

). However, for a p-rep ρ ∈ C(Wp/q), we have

u2 = 1 (and s2 6= 1) and hence the projection is 2-to-1. Each conjugacy class of p-reps

has therefore precisely two representatives in P.

The principal idea to determine P is now that we can in fact restrict our attention

to the eigenvalues of the generator images, by working in the eigenvalue variety of the

Whitehead link (cf. section 1.5.3). The fact that we do not lose any information in taking

this “detour” stems from the existence of a birational equivalence between C(W ) and

the subvariety E0(W ) := E(W )− Er(W ), where E(W ) ⊂ (C∗)4 is the eigenvalue variety,

Er its component arising from reducible representations, and the overline denotes the

Zariski closure. We choose coordinates (s, t, u, v) ∈ E(W ) given by the upper left entries

of ρ(µ0), ρ(λ0), ρ(µ1) and ρ(λ1), respectively. An easy computation shows that Er =

{t = v = 1} with respect to this choice of coordinates. See section 1.5.3 for details on

the definition of the eigenvalue variety and the birational equivalence.

The situation is summarized in the following diagram of varieties and their subsets

corresponding to P:

C(Wp/q)⊂

// C(W ) ' // E0(W )

P

OO

∼= // P′

OO

∼= // P′′

OO(2.5)

Here P′ is the set of partial p-reps ρ′ ∈ C(W ) induced by the p-reps ρ ∈ C(Wp/q). Define

the set P′′ := e(P′) to be the image of P′ under the birational equivalence to E0. That

is, a partial p-rep ρ′ will be sent to (s, t, u, v) ∈ E0(W ) satisfying sptq = 1 and u2 = 1.

This means that the set P′′ lies in the intersection F of the following subvarieties

of E0(W ):

P′′ ( F := E0(W ) ∩ {sptq = 1} ∩ {u2 = 1}. (2.6)

34

It is important to note that P′′ is a proper subset of F, as there are indeed points

in E0(W ) ∩ {sptq = 1} corresponding to representations which do not factor through

π1(Wp/q).

The advantage of this description now lies in the fact that we can first intersect

E0(W ) with the set {u2 = 1} (which will significantly simplify the defining polynomials

involved), and then intersect the resulting set with the subset {sptq = 1} corresponding

to the Dehn filling.

2.2 The subset of the eigenvalue variety corresponding to p-reps

We start with the parametrization of E0(W ) by the following defining set of

polynomials in the variables s, t, u and v (cf. section 1.5.3):

h1 = t− s2t+ s2t2 − s4t2 − u2 − 2s2tu2 + s4tu2 − t2u2 + 2s2t2u2

+ s4t3u2 + tu4 − s2tu4 + s2t2u4 − s4t2u4,

(2.7a)

h2 = s2 − v − s4v + u2v + 2s2u2v + s4u2v − s2u4v + s2v2 − u2v2

− 2s2u2v2 − s4u2v2 + u4v2 + s4u4v2 − s2u4v3,

(2.7b)

h3 = s4t− s6t− s2tu2 + s4tu2 + s6t2u2 − s2u2v

+ u4v + s2u4v − 2s4tu4v − u6v + s2u6v2.

(2.7c)

Intersecting with the subvariety given by {u2 = 1} then simplifies the defining polyno-

mials to

g1 = (t− 1)[t(t− 1)s4 + 4ts2 + (1− t)

], (2.8a)

g2 = s2(1− v)(v + 1)2, (2.8b)

g3 = s2[t(t− 1)s4 + 2t(1− v)s2 + (v2 − t)

]. (2.8c)

Since s 6= 0, it can be easily verified that the subvariety defined by these polynomials

splits into the following two irreducible components:

E0(W ) ∩ {u2 = 1} =

{u2 = 1, t = v = 1} ∪ {u2 = 1, v = −1, s4t2 + (−s4 + 4s2 − 1)t+ 1 = 0}(2.9)

35

Note that the first component corresponds to reducible representations.

Finally, filling the first cusp with the slope p/q now corresponds to

sptq = 1, (2.10)

and so we obtain the set F = E0(W ) ∩ {u2 = 1} ∩ {sptq = 1} as

F = Fr ∪ Fi = {u2 = t = v = 1, sp = 1}

∪ {u2 = 1, v = −1, s4t2 + (−s4 + 4s2 − 1)t+ 1 = 0, sptq = 1}.(2.11)

Let us denote the reducible and irreducible parts of P′′ by (P′′)r ( Fr and (P′′)i (

Fi, respectively, so that P′′ = (P′′)r ∪ (P′′)i.

2.3 Solutions corresponding to reducible representations

From equation (2.11), we can immediately read off the solutions corresponding to

reducible representations as

Fr = {(s, 1,±1, 1) | sp = 1} , (2.12)

i.e. the eigenvalue s is necessarily a |p|-th root of unity.

We now need to determine which of these solutions also correspond to p-reps

ρ ∈ R(Wp/q), or equivalently to partial p-reps ρ′ ∈ R(W ).

The reader can easily verify that a reducible representation ρ′ ∈ C(W ) corre-

sponding to the eigenvalues (±1, 1,±1, 1) is necessarily abelian6 and hence not a partial

p-rep. On the other hand, all points (s, 1,±1, 1) ∈ Fr with s 6= ±1 give indeed rise to

partial p-reps: Since u2 = t = v = 1 implies ρ′(λ0) = I, we have ρ′(µp0λq0) = ρ′(µp0) = I

and hence ρ′ factors through π1(Wp/q). Furthermore, s2 6= 1 implies that ρ′ is indeed

non-abelian.

6The conditions f(s, u, c) = 0 (cf. equation (1.25)) and t = v = 1 (cf. equations (1.32)) imply

that c = 0 in the normal form (2.4).

36

This means that the subset (P′′)r ( Fr is given by

(P′′)r = {(s, 1,±1, 1) | sp = 1, s 6= ±1} . (2.13)

To obtain the cardinality of this set, first note that −1 is a |p|-th root of unity iff

p is even. Since we always have two choices for u = ±1, we therefore obtain

#(P′′)r =

2(|p| − 1) if p is odd,

2(|p| − 2) if p is even.(2.14)

Note that the reducible p-reps are therefore characterized solely by the possible

solutions for the eigenvalue s, which were obtained as the “non-trivial” complex roots

of a polynomial in one variable, sp = 1. We will see in the following section that this

holds also in the irreducible case. Whereas it was a triviality to obtain the roots in

the reducible case, we will see that this will be a much more delicate problem in the

irreducible case. The rest of this chapter is therefore dedicated to this issue. We will

give a summary of the results in section 2.5, postponing the detailed analysis of the

roots to section 2.6.

2.4 Solutions corresponding to irreducible representations

Let us denote the defining polynomials of Fi by

k1 = sptq − 1, (2.15)

k2 = s4t2 + (−s4 + 4s2 − 1)t+ 1, (2.16)

so that Fi = {u2 = 1, v = −1, k1 = k2 = 0}.

To determine the solutions of k1 = k2 = 0 we will use elimination theory (see

section 1.3 and for example [CLO97]), and in particular the resultant of k1 and k2 with

37

respect to t:

resp,q := Resultant(k1, k2, t)

= det

sp 0 s4

0 sp −s4 + 4s2 − 1 s4

0 1 −s4 + 4s2 − 1

1. . .

......

. . .

. . . s4

−1 0 −s4 + 4s2 − 1

0 −1 1

(q+2)×(q+2)

= sp+2q(sp−2q + (−1)q+12Tq(−s2 + 2− s−2) + s−p+2q

)= s2p − sp+4q + · · · − sp + s4q.

(2.17)

See section 2.6 for the details of the calculation of this resultant, the definition and

properties of the Chebyshev polynomials Tq and a detailed analysis of its roots and

symmetries. First note that resp,q is in fact a Laurent polynomial7 in Z[s±1] when

p < 0. The middle terms are given recursively with respect to q and can be expressed

by the Chebyshev polynomial of first type Tq(y) by using the variable substitution

y = −s2 +2−s−2. Since Tq(y) is a polynomial of degree q in y, this gives a symmetrical

Laurent polynomial of span 4q in the variable s. Note that it only contains monomials

of even exponent, contributing precisely the terms between sp+4q and sp in resp,q.

For us, the main importance of this resultant will be that its roots characterize

the solutions of the system {k1 = k2 = 0}:

The extension theorem of elimination theory (cf. [CLO97, §1, Theorem 3] and

section 1.3) shows that the system {k1 = 0, k2 = 0} has a solution (s, t) if and only if

s 6= 0 is a root of this resultant. We will call s a partial solution in this case.

Furthermore note that given a partial solution s, the value for t is uniquely

determined by k1 = 0 and k2 = 0: To see this, note that k2 = 0 implies t2 =

7We will often use the word polynomial to also include Laurent polynomials.

38

−s−4((−s4 + 4s2 − 1)t+ 1

), and hence higher powers of t can recursively be replaced

by linear terms in t. The equation k1 = sptq − 1 = 0 can therefore always be expanded

to an equation which is linear in t.

Denote the unique solution for t for a given s by ts — each root s 6= 0 of the

resultant therefore corresponds to two eigenvalue points (s, ts,±1, 1) in Fi ⊂ E0(W ), so

that we have

Fi = {(s, ts,±1, 1) | resp,q(s) = 0}. (2.18)

As in the reducible case, not all of these points correspond to representations

which will factor through π1(Wp/q) to give p-reps in R(Wp/q). This is due to the fact

that the eigenvalue condition sptq = 1 does not necessarily imply that ρ(µ0)pρ(λ0)q = I.

In fact, we have:

Proposition 2.4.1 The roots s /∈ {0, 1,−1} of resp,q correspond to irreducible p-reps

of π1(Wp/q); the roots s = ±1 (when they arise) correspond to discrete faithful repre-

sentations of π1(W ) and which therefore do not factor through π1(Wp/q).

Proof. For s = ±1, a direct calculation shows that an irreducible representation with

s2 = u2 = 1 is conjugate to the form

ρ(µ0) =

s −su± i

0 s

, ρ(µ1) =

u 0

1 u

, and ρ(λ0) =

−1 4su

0 −1

, (2.19)

where s, u ∈ {±1}. (These are in fact, up to conjugation, all discrete faithful representa-

tions into SL2(C) corresponding to the unique complete hyperbolic structure of the un-

filled Whitehead link complement.) For each choice of s = ±1, there are four such repre-

sentations. As the subgroup generated by the parabolic elements ρ(µ0), ρ(λ0) ∈ SL2(C)

is therefore isomorphic to Z2, the equation ρ(µ0)pρ(λ0)q = I can only be satisfied by

p = q = 0 (which is not a valid filling slope).

For s 6= ±1, we first note that the inverse map of the birational equivalence

between E0(W ) and C(W ) (cf. equation (1.35) in section 1.5.3)

e−1 : E0(W ) → C(W ), (s, t, u, v) 7→(s, u,

s2(t− 1) + u2(1− v)s−1u−1(s2 − u2)

)(2.20)

39

is well-defined (since u2 = 1) and has degree one. The two points (s, ts,±1,−1) as-

sociated to a root s therefore correspond to two partial p-reps in C(W ). To see that

these indeed factor through π1(Wp/q), note that for s 6= ±1, both ρ(µ0) and ρ(λ0) are

simultaneously diagonizable, since they are not parabolic and commute. The condition

sptq = 1 then immediately implies ρ(µ0)pρ(λ0)q = I. �

2.4.1 Roots of the resultant

We state a few facts about the roots of the resultant resp,q, which will be proved

in detail in section 2.6. First we claim that ±1 is a root of the resultant in the following

situations:

• If q is even (and hence p odd), +1 is a root of order 2.

• If both q and p are odd, −1 is a root of order 2.

In the remaining case (q odd and p even), neither of ±1 is a root. This can be directly

verified from the equations k1 = k2 = 0 (cf. equations (2.15) and (2.16)): If s = ±1,

equation k2 = 0 necessarily implies ts = −1. Using k1 = 0, we then immediately see

that (1,−1) is a solution iff q is even, and that (−1,−1) is a solution iff p + q is even

(which — under the assumption that p and q are coprime — is equivalent to both p and

q being odd). We refer to section 2.6.3 for the proof that the order of ±1 is equal to 2

(when it arises as a root).

We will see in section 2.6.3 that for odd p, all remaining roots are distinct and

of order 1. This follows in fact from the results of chapter 4, using the connection to

the minimal Culler-Shalen seminorm which we will examine in chapter 3. It is to be

expected that the corresponding calculation of the Culler-Shalen seminorm for even p

(which we do not cover in this thesis) will show the simplicity of the roots for even p.

At this moment, we can confirm this by analytical methods when q = 1.

Examples of the typical distribution of the roots in the two cases p > 4q > 0 and

0 < p < 4q can be seen on page 48 in figures 2.1(a) and 2.1(b), respectively.

40

We will now determine the number of non-zero roots of the resultant. We will

assume that p and q are coprime integers with q > 0 and p ∈ Z arbitrary. Consider

the resultant as a Laurent polynomial in Z[s±1]. The number of non-zero roots (with

multiplicities) is then given by its span, i.e. the difference of the maximal and minimal

degrees8, which can be expressed as max(2p, p+ 4q, 4q)−min(p, 4q, 2p). Dependent on

p and q, we therefore get the numbers shown in table 2.1.

p/q ∈ (−∞, 0) (0, 4) (4,∞)

# of non–zero solutions 2|p|+ 4|q| 4|q| 2|p| − 4|q|

Table 2.1 Number of non–zero solutions (including multiplicities) of the resultant

The special cases p/q ∈ {0, 4} have to be dealt with separately: For p/q = 0,

we get res0,1 = 4s6, whereas for p/q = 4, we get res4,1 = 4s2 — in both cases, there

are no non-zero solutions of the resultant and hence also no solutions of the system

k1 = k2 = 0. Note that this complete lack of irreducible p-reps for these manifolds can

be seen as a (strong) confirmation of the fact that W0 and W4 are toroidal manifolds

which hence do not admit a complete hyperbolic structure.

Discarding 0, 1 and −1 from the roots of resp,q, we therefore obtain an upper

bound for the number of distinct roots as given in table 2.2. In fact, we will conjecture

in section 2.6.3 (cf. Conjecture 2.6.8) that all roots apart from 0 and ±1 are simple

roots, and that hence the bounds in table 2.2 are indeed the exact numbers of distinct

roots.

p/q ∈ (−∞, 0) (0, 4) (4,∞)

p odd 2|p|+ 4|q| − 2 4|q| − 2 2|p| − 4|q| − 2

p even 2|p|+ 4|q| 4|q| 2|p| − 4|q|

Table 2.2 Upper bound for the number of distinct roots s 6∈ {0,±1} of resp,q

8These degrees are more easily deduced by using a normalized version of the resultant, cf.

section 2.6

41

2.4.2 Symmetries

Note that expressed in the symmetrical form

resp,q = sp+2q(sp−2q + (−1)q+12Tq(−s−2 + 2− s2) + s−p+2q

), (2.21)

the resultant is (up to a monomial factor) symmetric with respect to inversion of s (see

section 2.6 for the details). It follows that for each partial solution s, its inverse s−1 is

also a partial solution. Furthermore note that we can write k2 in the form

k2 = s2t(s2t+ s−2t−1 + 4− s2 − s−2

). (2.22)

From this, we easily see that (s, ts) is a solution of k2 = 0 iff (s−1, t−1s ) is a solution.

Furthermore, (s−1, t−1s ) clearly satisfies k1 = 0 iff (s, ts) does.

2.5 The number of conjugacy classes of p-reps

We will now put together the results of the previous sections to obtain the exact

count of conjugacy classes of p-reps for the manifold Wp/q.

Recall that we have determined the set of eigenvalue solutions F = Fr∪Fi ⊂ E(W ),

where

Fr = {u2 = t = v = 1, sp = 1}, (2.23)

Fi = {u2 = 1, v = −1, resp,q(s) = 0, t = ts}. (2.24)

As noted before, the set P′′ ⊂ E0(W ) is a strict subset of F, as we need to exclude

s = ±1 from both the reducible and irreducible case: In the reducible case s = ±1

corresponds to abelian representations, and in the irreducible case to representations

which do not factor through π1(Wp/q).

More specifically, we have

(P′′)r = {(s,±1, 1, 1) | sp = 1, s 6= ±1}, (2.25)

(P′′)i = {(s,±1, ts,−1) | resp,q(s) = 0, s 6= ±1}. (2.26)

42

Recall that we already determined the cardinality of the set (P′′)r in (2.14). For

the irreducible count, we saw that +1 (or −1, respectively) is a root of order 2 of resp,q

iff q is even and p is odd (both p and q are odd, respectively). Using the upper bound9

for the number of distinct roots of resp,q from table 2.2, we therefore obtain the upper

bound for the cardinality of P′′ shown in table 2.3. Taking the sum gives us the upper

bound for the cardinality of P′′, shown in table 2.4. Now recall that by construction,

the cardinality of P′′ is equal to the cardinality of P′ and that of P. Furthermore, we

saw that each conjugacy class of p-reps in C(Wp/q) is represented by two elements in

C(Wp/q) (corresponding to the pair of eigenvalue points (s, t, u, v) and (s−1, t−1, u, v)).

To obtain the number of conjugacy classes of p-reps, we therefore have to divide by 2.

Since the two representatives for a conjugacy class in C(Wp/q) clearly have the same

character independent of whether they are reducible or irreducible, we see that this

number also coincides with the number of characters of p-reps. The final results are

summarized in table 2.5.

Though we will only make use of the upper bound for the number of conjugacy

classes as obtained here, we remark that this bound is in fact attained whenever Con-

jecture 2.6.8 is known to be true. The results of chapters 3 and 4 will in fact show that

this is the case when p is odd — and it is to be expected that this is true in general.

p/q ∈ (−∞, 0) (0, 4) (4,∞)

p odd 4|p|+ 8|q| − 4 8|q| − 4 4|p| − 8|q| − 4

p even 4|p|+ 8|q| 8|q| 4|p| − 8|q|

Table 2.3 Upper bound for the number of elements in (P′′)i

p/q ∈ (−∞, 0) (0, 4) (4,∞)

p odd 6|p|+ 8|q| − 6 2|p|+ 8|q| − 6 6|p| − 8|q| − 6

p even 6|p|+ 8|q| − 4 2|p|+ 8|q| − 4 6|p| − 8|q| − 4

Table 2.4 Upper bound for the number of elements in P′′

9Which is conjecturally attained

43

p/q ∈ (−∞, 0) (0, 4) (4,∞)

p odd 3|p|+ 4|q| − 3 |p|+ 4|q| − 3 3|p| − 4|q| − 3

p even 3|p|+ 4|q| − 2 |p|+ 4|q| − 2 3|p| − 4|q| − 2

Table 2.5 Upper bound for the number of conjugacy classes of SL2(C) p-reps of

π1(Wp/q)

2.6 The eigenvalue resultant and its properties

We give the details for the calculation of the resultant from section 2.4. Since

we will only be interested in non-zero roots, we will use the term “polynomial” also for

Laurent polynomials, if not otherwise noted.

We start by introducing a series of polynomials which will appear later as minors

in the calculation of the resultant. For brevity, let σ := −s4 + 4s2 − 1. Furthermore,

define the following k × k matrix, for 3 ≤ k ≤ q:

Qk :=

σ s4

1 σ

1. . .. . . s4

. . . σ s4

1 σ

k×k

(2.27)

For k ≤ 2, we set:

Q2 :=

σ s4

1 σ

, Q1 :=(σ), Q0 :=

(1)

and Qk :=(0)

for k ≤ 0.

(2.28)

To simplify notation, let gk := detQk for k ∈ Z. In particular, we have gk = 0 for k < 0,

g0 = 1 and g1 = σ.

With this notation in place, we are now ready to calculate the resultant of k1 and

k2 with respect to the variable t as the determinant of the associated Sylvester matrix

44

(cf. section 1.3.3):

resp,q : = Resultant(sptq − 1, s4t2 + σt+ 1, t)

= det

sp 0 s4

0 sp σ s4

0 1 σ

1. . .

......

. . . s4

. . . σ s4

−1 0 1 σ

0 −1 1

(q+2)×(q+2)

(2.29)

Consecutively applying Laplace development along the first two columns, we obtain

resp,q = s2p + (−1)q+3sp detQq

+ (−1)q+3(−sp) det

s4 0

1 σ s4

1 σ. . .

1. . . s4

. . . σ s4

0 1

q×q

+ (−1)2q+6 det

s4

σ s4

1 σ. . .

1. . . s4

. . . σ s4

1 σ s4

q×q

= s2p + (−1)q+1sp(gq − s4gq−2

)+ s4q

(2.30)

Note that we assumed q > 0 and that we can calculate this resultant even for p < 0

by considering the Laurent polynomial sptq − 1 as a polynomial in t with coefficients in

Z[s±1].

To better understand the structure of resp,q, we will now investigate the polyno-

mials gk. First note that we have the following recursive formula, which can be obtained

45

by developing the matrix Qk along the first column:

gk = g1gk−1 − s4gk−2 for k ∈ Z. (2.31)

It will be useful to write these polynomials in a more symmetric form as Laurent

polynomials:

hk := s−2kgk (2.32)

so that we get

h0 = 1, hk = 0 for k < 0, (2.33)

h1 = −s2 + 4− s−2 and (2.34)

hk = h1hk−1 − hk−2 for k ∈ Z. (2.35)

Note that after introducing a change of variable y(s) := h1/2, equation (2.35) can be

recognized to be precisely the recursion formula

hk = 2yhk−1 − hk−2 (2.36)

satisfied by the Chebyshev polynomials of the first and second kind, Tk and Uk re-

spectively, where T0(y) = 1, T1(y) = y, U0(y) = 1, U1(y) = 2y. Furthermore define

Tk = Uk = 0 for k < 0. Since by definition h0 = 1 and h1 = 2y, we find that in fact

hk(s) = Uk(y(s)). (2.37)

Since the properties of these classical orthogonal polynomials will play an impor-

tant role in our investigation, we summarize a few classical results (cf. [Sze75]):

Proposition 2.6.1 (Properties of the Chebyshev polynomials)

1. Tk(y) = 0 (k < 0), T0(y) = 1, T1(y) = y, Tk(y) = 2yTk−1(y)− Tk−2(y).

2. Uk(y) = 0 (k < 0), U0(y) = 1, U1(y) = 2y, Uk(y) = 2yUk−1(y)− Uk−2(y).

3. Letting y = cosx, the following trigonometric representations can be used where

they are defined: Tk(y) = cos(kx), Uk(y) = sin(k+1)xsinx .

46

4. For k > 0, the roots of Tk(y) are yj,1 = cos (2j−1)π2k (j = 1, . . . , k), and those of

Uk(y) are yj,2 = cos jπk+1 (j = 1, . . . , k). In particular, all roots lie in the open

interval (−1, 1) and are simple and distinct, and are also distinct from all roots of

the other polynomials in the series.

5. Tk(y) = Uk(y)− yUk−1(y) = 12 (Uk(y)− Uk−2(y)) (k ≥ 1).

6. ddyTk(y) = kUk−1(y).

7. Tk(±1) = (±1)k, Uk(±1) = (±1)k(k + 1).

8. (1 − y2)T ′′k − yT ′k + k2Tk = 0 and (1 − y2)U ′′k − 3yU ′k + k(k + 2)Uk = 0 (special

cases of the Sturm-Liouville differential equation).

9. Tk(y)− Tk−2(y) = −2(1− y2)Uk−2(y) (k ≥ 2).

Proof. See standard literature on orthogonal polynomials, e.g. [Sze75]. For (9) use

induction or the trigonometric form (3) and the addition theorem for cos. �

This immediately implies the following for the polynomials hk(s):

Corollary 2.6.2

1. hk is a Laurent polynomial of span 4k containing only even powers of s. In par-

ticular, hk(−s) = hk(s).

2. The 4k roots of hk(s) are real, simple and distinct.

3. hk(±1) = Uk(1) = k + 1.

4. hk(±1±√

2) = Uk(−1) = (−1)k(k + 1)

Proof. For (1), observe that hk = s−2kgk = s−2k detQk where Qk only contains even

powers of s. For (2), solving y = 12h1(s) = 1

2(−s2 + 4− s−2) for s gives

s = ±√

2− y ±√

(y − 1)(y − 3)

47

(where the signs can be chosen independently). It is easy to see that these four values

are real iff y ≤ 1. This is certainly satisfied if y is a root of Uk by Proposition 2.6.1(4).

Since in this case y ∈ (−1, 1), it is easy to see that the four values for s are distinct.

In total, we therefore have 4k distinct, real roots si,j (i = 1, . . . , 4 and j = 1, . . . , k) s.t.

hk(si,j) = Uk(yi) = 0. For (3) and (4), note that y(±1) = 1 and y(±1±√

2) = −1. �

Using the polynomials hk, we now can rewrite the resultant as

resp,q(s) = s2p + (−1)q+1sp(gq − s4gq−2) + s4q

= s2p + (−1)q+1sp+2q(hq − hq−2) + s4q

= sp+2q(sp−2q + (−1)q+1(hq − hq−2) + s−p+2q

) (2.38)

Using hq(s)− hq−2(s) = Uq(y)− Uq−2(y) = 2Tq(y) by 2.6.2 (5), we get

resp,q = sp+2q(sp−2q + (−1)q+12Tq(y) + s−p+2q

)(2.39)

Since we are only interested in non-zero roots, we define a normalized version of

the resultant asnresp,q := s−(p+2q) resp,q

= sp−2q + (−1)q+12Tq(y) + s−p+2q.(2.40)

Since Tq(y) is a polynomial of degree q in y, and y is a symmetrical Laurent polynomial

of span 4 in s, it follows that Tq(y) is a symmetrical Laurent polynomial of span 4q in

s. From this we immediately see that the span of nresp,q (and therefore the number of

non-zero roots of resp,q) is given by 2 max(|p− 2q|, 2q).

The distribution of the roots falls into the two distinct cases p > 4q > 0 (or

p < 0 < 4q) and 0 < p < 4q. We show examples of the typical distribution in figures

2.1(a) and 2.1(b), respectively.

Note that although the roots converge towards the unit circle for p/q → ∞ and

seem distributed very regularly with respect to their arguments, the same is not true

when p/q is near 4, as can be seen in the example in figure 2.2 for res65,16.

48

(a) res66,3 (b) res66,23

Figure 2.1 The generic distribution of the non-zero roots of resp,q

Figure 2.2 The distribution of the non-zero roots of res65,16

49

The two borderline cases with respect to the distribution behaviour are when

p/q = 0/1 and p/q = 4/1 — in these cases, resp,q has no non-zero roots. The case

p/q = 2/1 is also significant, as it is the case when there are precisely four roots in total,

all of which are real-valued (cf. Proposition 2.6.4). Furthermore, note that the resultant

is symmetric in p with respect to this case (i.e. p = 2q), as we have:

nres−p+4q,q = s(−p+4q)−2q + (−1)q+12Tq(y) + s(p−4q)+2q = nresp,q . (2.41)

It therefore suffices to study the roots of the resultants for p ≥ 2q > 0.

The symmetries appearing in the distribution of the roots for fixed p and q will

be studied in the next section.

2.6.1 Symmetries of the roots

The Laurent polynomials nresp,q satisfy the following symmetries:

Proposition 2.6.3

1. nresp,q(s−1) = nresp,q(s)

2. nresp,q(s) = 0 ⇐⇒ nresp,q(s) = 0

3. nresp,q(s) = 0 = nresp,q(−s) iff p even.

Proof. For (1), observe that y(s−1) = y(s). For (2), note that nresp,q is a Laurent

polynomial with real (in fact integer) coefficients. For (3), first note that y(−s) = y(s)

and hence

nresp,q(−s)− nresp,q(s) = ((−1)p − 1)(sp−2q + s−p+2q

).

This is clearly 0 when p is even. Conversely, assume this difference to be 0 but p to

be even. Then s2(p−2q) = −1, and in particular s = eiφ for some 0 ≤ φ < 2π. Then

nresp,q(s) = (−1)q+12Tq(y(s)) = (−1)q+12Tq(2 − cos(2φ)) 6= 0, since the roots of Tq(y)

lie in the interval (0, 1) by Proposition 2.6.1 (4). �

50

In particular, this implies that the roots of nresp,q (and hence of resp,q) come

generically in quadruples: s, s−1, s, s−1. Furthermore, if p is even, the symmetry is

generically 8-fold, adding −s, −s, −s−1 and −s−1.

2.6.2 Real and imaginary roots

We now study the question whether any roots apart from ±1 of resp,q can be real

or pure imaginary.

Proposition 2.6.4 When p > 4q > 0 or p < 0, the only possible real-valued roots of

resp,q are ±1. When 0 < p < 4q and p is odd, there are two additional real-valued,

positive roots. When 0 < p < 4q and p is even (and hence q odd), there are precisely

four real-valued roots in total, none of which is ±1.

Proof. Assume resp,q(s) = nresp,q(s) = 0 with s > 0. Then

sp−2q + s−(p−2q) = (−1)q2Tq(y(s)).

Denote the left-hand side of this equation by φ(s), the right-hand side by ψ(s). We

will study intersections of these two real-valued functions. Note that φ(s) is strictly

monotonously decreasing for 0 < s < 1, has a global minimum of value 2 at s = 1,

and is strictly monotonously increasing for s > 1. Furthermore, note that the range

of y(s) is (−∞, 1), with global maximum at s = 1. This implies that the range of

ψ(s) = (−1)q2Tq(y(s)) is (−∞, 2), using the fact that Tq(y) is a polynomial of degree q

in y, whose zeros and relative extrema of value±1 are all contained in the interval [−1, 1].

More specifically, ψ oscillates with values between −2 and 2 for s ∈ [1 −√

2, 1 +√

2]

(with ψ(1) = (−1)q2) and tends strictly monotonously to +∞ for s→ 0 and s→∞.

Apart from the possibility of an intersection at s = 1 (in the case that q is

even), other intersections between φ and ψ can therefore only occur outside the interval

(1 −√

2, 1 +√

2) when the maximal degree in s of ψ is larger than that of φ. But

maxdegs φ = |p − 2q| = −mindegs φ and maxdegs ψ = 2q = −mindegs ψ. It is then

easy to see that for p > 4q > 0 or p < 0, maxdegs ψ < maxdegs φ and there are no such

51

intersections. The same argument then shows that for 0 < p < 4q, there are precisely

two positive real-valued roots different from 1 (both being inverses of each other, by

Proposition 2.6.3 (1).

To analyze the remaining case s < 0, note that φ(−s) = (−1)pφ(s) while ψ(−s) =

ψ(s). If p is odd, this means that there cannot be any intersection apart from possibly

s = −1 (which arises if q is odd), as the two functions diverge in opposite directions. If

p is even (and hence q odd), we get as roots the negatives of the roots found for s > 0.

In total, there are either 0 or 4 real roots in this case, as resp,q(±1) 6= 0. �

Note that this includes the case p/q = 2/1, when there are only four non-zero

roots in total, all of which are real and distinct from ±1.

Proposition 2.6.5 When p > 4q > 0 or p < 0, resp,q has imaginary roots iff p ∈ 4Z.

In that case, there are four such roots. When 0 < p < 4q, there are no imaginary roots.

Proof. Similar to the proof of Proposition 2.6.4. Assume resp,q(ix) = nresp,q(ix) = 0

with x > 0. Then

i−p+2q((−1)pxp−2q + x−p+2q

)= (−1)q2Tq

(x2+4+x−2

2

).

Now note that the right-hand side is always real-valued, while the same for the left-hand

side is only true when p is even. In that case, the equation is equivalent to

φ(x) := (−1)p/2(xp−2q + x−p+2q

)= 2Tq

(x2+4+x−2

2

)=: ψ(x).

As ψ(x) > 0 for all x ∈ R, this equation can only be satisfied if p/2 is even, i.e. if

p ∈ 4Z. Under this assumption, comparison of the maximal degrees then shows that

there is no intersection of φ and ψ when 0 < p < 4q, and precisely two intersections

(which are inverses of each other) when p > 4q > 0 or p < 0. The symmetry with

respect to complex conjugation then shows that there are in total four pure imaginary

roots in this case, and none when p /∈ 4Z. �

We also remark the following:

52

Lemma 2.6.6 None of the roots s /∈ {±1} of resp,q lies on the unit circle.

Proof. Simply note that

nresp,q(eiφ) = 2(cos(p− 2q)φ+ (−1)q+1 Tq(2− cos 2φ)︸ ︷︷ ︸≥1

). �

2.6.3 Multiple roots

In this section, we will discuss which of the roots of resp,q and nresp,q are multiple

roots.

An easy calculation (using Proposition 2.6.1) shows that the first and second

derivatives of nresp,q are given by

nres′p,q = (p− 2q)(sp−2q−1 − s−p+2q−1

)+ (−1)q+1qUq−1(y)h′1(s) (2.42)

and

nres′′p,q = (p− 2q)((p− 2q − 1)sp−2q−2 + (−p+ 2q − 1)s−p+2q−2

)+(−1)q+1q

(h′′1(s)Uq−1(y) + 1

2(h′1(s))2 d

dyUq−1(y)),

(2.43)

where

h′1(s) = −2(s− s−3) and h′′1(s) = −2(1 + 3s−4). (2.44)

We can now directly verify that

nresp,q(1) = 2(1 + (−1)q+1

)=

0 if q is even (and hence p odd),

4 if q is odd

(2.45)

andnresp,q(−1) = 2(−1)p

(1 + (−1)p+q+1

)=

0 if p+ q is even (i.e. p and q odd),

4(−1)p if p+ q is odd.

(2.46)

53

Furthermore, we have:

nres′p,q(±1) = (−1)q+1qUq−1(+1)h′1(±1)︸ ︷︷ ︸0

= 0 (2.47)

and

nres′′p,q(±1) = 2(p− 2q)2(±1)p + 8q2(−1)q. (2.48)

From this it follows that nres′′p,q(±1) 6= 0 unless p/q = 0/1 or p/q = 4/1 (which are

precisely the cases when resp,q has no non-zero roots). This implies that also res′′p,q(±1) 6=

0, and hence that ±1 are indeed of order 2 when they arise as roots. We summarize the

result:

Lemma 2.6.7 When q is even (and hence p odd), +1 is a root of order 2 of resp,q.

When both p and q are odd, −1 is a root of order 2. When p is even (and q odd), none

of ±1 is a root. �

The remaining part of this section is devoted to discussing the validity of the

following conjecture:

Conjecture 2.6.8 Apart from ±1, all non-zero roots of resp,q are simple. The

number of distinct roots s /∈ {0,±1} is therefore 2 max(|p − 2q|, 2q) − 2 when p is

odd, and 2 max(|p− 2q|, 2q) when p is even.

Due to the recursive nature of the polynomials nresp,q, it has proven difficult to

approach this conjecture with analytical methods. At the moment of this writing, we

are able to give an analytical proof only when q = 1. On the other hand, we are able to

prove the conjecture when p is odd in all generality through topological methods, via

the computation of the total Culler-Shalen seminorm in chapter 4.

We will first discuss the analytical method.

54

2.6.3.1 An analytical approach to the conjecture

Our main tool will be the fact that the first derivative of the symmetrized resultant

nresp,q inherits the symmetry with respect to the inversion of roots:

Lemma 2.6.9 nres′p,q(s−1

)= −s2 nres′p,q (s) and hence nres′p,q(s) = 0 if and only if

nres′p,q(s−1) = 0.

Proof. Simply note that y(s) = −s2 + 2− s−2 is symmetric with respect to inversion

and that h′1(s−1) = −s2h′1(s). �

On the other hand, for the non-normalized resultant, we have

res′p,q = 2ps2p−1 + 4qs4q−1

+ (−1)q+1sp+2q(2s−1(p+ 2q)Tq(y) + qUq−1(y)h′1(s)

),

(2.49)

and experimental evidence strongly indicates that the roots of res′p,q are not symmetric

with respect to inversion:

Conjecture 2.6.10 If res′p,q(s) = 0 = res′p,q(s−1), then s = ±1.

Before we discuss the different avenues undertaken towards a proof of this conjecture,

we will show how it implies the main conjecture:

Proof of Conjecture 2.6.8 (assuming Conjecture 2.6.10).

Let s be a multiple non-zero root of resp,q, so that resp,q(s) = res′p,q(s) = 0. Since the

orders of roots of resp,q and nresp,q coincide, we also have nres′p,q(s) = 0. But this implies

nres′p,q(s−1) = 0 by Lemma 2.6.9, and therefore also res′p,q(s

−1) = 0. But this implies

s = ±1 by Conjecture 2.6.10. �

Restricting ourselves to the case when q = 1, we are able to prove conjecture 2.6.10

to be true:

Proof of Conjecture 2.6.10 for q = 1.

We have

resp,1(s) = s2p−4 + 2sp−2 T1(y)︸ ︷︷ ︸y

+1 = s2p−4 − sp + 4sp−2 − sp−4 + 1, (2.50)

55

from which we obtain

res′p,1(s) = sp−5((2p− 4)sp − ps4 + 4(p− 2)s2 − (p− 4)

). (2.51)

Let now s be a multiple root of resp,q, i.e. resp,q(s) = res′p,q(s) = 0. Dividing by sp−5,

we solve (2.51) for the term with sp and obtain (2p−4)sp =(ps4−4(p−2)s2 +(p−4)

).

We now also assume s−1 to be a root, i.e. res′p,1(s−1) = 0. Solving the equivalent

of equation (2.51) for (2p − 4)s−p and multiplying by the expression for (2p − 4)sp

obtained earlier, we get the condition

s−4(s− 1)2(s+ 1)2(p(p− 4)s4 + (−6p2 + 24p− 32)s2 + p(p− 4)

)︸ ︷︷ ︸=:ψ

= 0. (2.52)

We will show that the factor ψ cannot be zero10, and hence conclude that s = ±1.

Note that the four roots of ψ for s in terms of p are

±

((3(p− 2)2 + 4

)±

(2|p− 2|

√2(p− 2)2 + 8

))1/2

√p(p− 4)

(2.53)

But in fact these roots are all real-valued when p > 4 or when p < 0, and all imaginary

when 0 < p < 4. To see this, simply observe that

(3(p− 2)2 + 4

)2 −(2|p− 2|

√2(p− 2)2 + 8

)2= 4p2(p− 4)2 ≥ 0 (2.54)

which shows that the numerator in (2.53) is always real-valued.

But we know that for p > 4 or p < 0, the only possibilities for real-valued, non-

zero roots of resp,1 are ±1 (cf. Proposition 2.6.4). Similarly, Proposition 2.6.5 shows

that for 0 < p < 4 none of the non-zero roots of resp,1 are pure imaginary. So ψ in fact

does not share any roots with resp,1. This concludes the proof of Conjecture 2.6.10 (and

hence of Conjecture 2.6.8) in the case q = 1. �

10It is interesting to note that a more general version of ψ will appear again in section 3.2,

equation (3.42).

56

Let us now discuss the difficulties and possible approaches for proving Conjec-

ture 2.6.10 in the general case. Starting with the assumption res′p,q(s) = 0 = res′p,q(s−1),

one would like to deduce that s = ±1 is the only possibility. The main difficulty now

lies in the circumstance that starting from this assumption and trying to eliminate the

Tq(y) and Uq−1(y) terms always seems to result in the conclusion resp,q(s) = 0, which

then begs the original question which s ∈ C can satisfy resp,q(s) = 0 = res′p,q(s).

To avoid this kind of circular argument, one possible avenue seems to try to

determine the general symmetry of the roots of res′p,q without appealing to the fact that

resp,q(s) = 0. In this respect, a slightly different normalization seems to be even more

promising. Without loss of generality, assume p > 4q > 0 and let resp,q := s−4q resp,q.

We then get

res′p,q(s) = (2p− 4q)s2p−4q−1

+ (−1)q+12sp−2q−1((p− 2q)Tq(y) + (s−2 − s2)qUq−1(y)

).

(2.55)

Experiments now seem to indicate that all roots of this derivative which lie strictly

inside the unit circle are in fact real-valued. If this were proven in general, the non-

symmetry of the roots with respect to inversion would follow immediately: Recall that

for p > 4q > 0 there are no real roots of resp,q apart from ±1 (cf. Proposition 2.6.4).

None of the non-real roots of res′p,q strictly outside of the unit circle could hence be roots

of resp,q. A very similar argument would show the conjecture in the case p < 0 < 4q (in

this case it seems that two roots of res′p,q outside the unit circle are pure-imaginary and

the rest are all real-valued). Unfortunately, the non-existence of non-real roots inside

(or outside, respectively) the unit circle also seems hard to show in general.

It is our belief that it should be possible to find an analytical proof for the general

case along similar ideas as the ones discussed above — but it still remains to be found

at the time of writing.

2.6.3.2 A topological approach to the conjecture

We will briefly describe how our results from chapters 3 and 4 give a proof of

Conjecture 2.6.8 when p is odd.

57

We will see in chapter 3 that the number of conjugacy classes of p-reps corresponds

precisely to the minimal total Culler-Shalen seminorm s. Allowing the possibility of

multiple roots of resp,q apart from 0 and ±1, we can only assume the upper bound for

the number of conjugacy classes (and therefore for s) given in table 2.5 (cf. page 43).

It now turns out that the knowledge of this upper bound is already sufficient to solve

certain linear systems corresponding to the total Culler-Shalen seminorm in section 4.3,

and thereby to completely determine s. We then find that the upper bound for the

number of conjugacy classes must in fact be attained. As a consequence, the upper

bound for the number of distinct roots of resp,q must be attained, and we conclude that

there are in fact no multiple non-zero roots apart from ±1.

As we do not cover the calculations of the total Culler-Shalen seminorm for even p

in this thesis, we can at this time make no statement about the truth of Conjecture 2.6.8

for p even and q > 1. On the other hand, it seems very probable that analogous

arguments to those in section 4.3 will hold in this case, and that our results can therefore

be extended to all p and q. This is also strongly supported by experimental evidence.

Chapter 3

The minimal non-zero value of the total Culler-Shalen

seminorm and p-reps

In this chapter, we determine the minimal Culler-Shalen seminorm ‖µ1‖ for the

manifolds Wp/q = Wrh(p/q, · ) in terms of the number of distinct roots of the polynomial

resp,q, using the results from chapter 2.

Recall from section 1.2 that the Culler-Shalen seminorm of an element α of the

peripheral subgroup im(π1(∂Wp/q) → π1(Wp/q)

)on a non-trivial1 curve Xi of the char-

acter variety X(Wp/q) is defined as

‖α‖Xi= deg fα, (3.1)

where fα : X0 → CP1 is defined on the smooth projective model Xi of Xi by fα =

(trace ρ(α))2 − 4.

Using the fact that Dehn filling along the meridian µ1 of Wp/q (which we view as

the meridian of the second boundary torus of the unfilled Whitehead link exterior) gives

a manifold with cyclic fundamental group, while µ1 is not a boundary slope, Corollary

1.1.4 of [CGLS87] shows that on each curve Xi ⊂ X(Wp/q), we have

‖µ1‖Xi= sXi , (3.2)

the associated minimal seminorm on Xi (cf. section 1.2).

1A curve containing the character of an irreducible representation.

59

By regarding the sum over all non-trivial curves Xi ⊂ Xirr(Wp/q), we obtain the

minimal total Culler-Shalen seminorm s := ‖µ1‖ =∑

i sXi .

Note that the degree of a function can be obtained by counting its zeros (or its

poles) with multiplicities, but this generally needs to be done in the smooth projec-

tive completion of its domain. We will discuss this issue in a moment, but will first

examine which characters χ ∈ X(Wp/q) are zeros of fµ1 . Since fµ1 = 0 is equivalent

to trace ρ(µ1) = ±2, we need to look for all non-abelian representations ρ ∈ R(Wp/q)

which send the meridian µ1 ∈ π1(Wp/q) to a parabolic2 element of SL2(C). This means

that the representations ρ to be considered are the SL2(C) p-reps of π1(Wp/q) which we

determined in chapter 2. In particular, we there determined the number of conjugacy

classes, and equivalently the number of characters, of p-reps, showing that this number

is bounded above by the numbers given in table 2.5.

The goal of this chapter is to show:

Theorem 3.0.11 The minimal total Culler-Shalen seminorm of the manifold Wp/q

is equal to the number of characters of p-reps of π1(Wp/q).

As a consequence, the upper bound given in table 2.5 also gives an upper bound

for the minimal Culler-Shalen seminorm. We summarize this in table 3.1.

p/q ∈ (−∞, 0) (0, 4) (4,∞)

p odd 3|p|+ 4|q| − 3 |p|+ 4|q| − 3 3|p| − 4|q| − 3

p even 3|p|+ 4|q| − 2 |p|+ 4|q| − 2 3|p| − 4|q| − 2

Table 3.1 Upper bound for the minimal Culler-Shalen seminorm s = ‖µ1‖ for

Wrh(p/q, · )

The proof of Theorem 3.0.11 will be obtained by showing:

2The possibility ρ(µ1) = ±I would imply that ρ were abelian.

60

1. None of the roots of fµ1 lie at ideal points of X(Wp/q).

2. Each character of a p-rep of π1(Wp/q) is a smooth point of X(Wp/q).

3. Each character of a p-rep of π1(Wp/q) is a simple zero of fµ1 .

The first claim is immediately clear, as it can be seen that fµ1 has a pole (and

hence no zero) at each ideal point: If fµ1 took a finite value at an ideal point, the fact

that π1(Wp/q(µ1)) is cyclic and that Wp/q is a small manifold3 unless p/q ∈ {0, 4} (cf.

the Appendix of [BGZ01] for a proof) would imply that µ1 was a boundary slope, a

contradiction (see e.g. the remarks after Corollary 9.2.7 in [Sha02]).

The other two claims are more subtle and will be addressed in the next sections: In

section 3.1 we will show that all characters of p-reps are smooth points of Xirr(Wp/q) (the

subvariety consisting of non-trivial curves in X(Wp/q)), using cohomology techniques.

We will then show in section 3.2 that each character of a p-rep is a simple zero of

fµ1 |Xirr(Wp/q).

3.1 Smoothness of the character variety at the characters of p-reps

In this section, we will show that each character of a p-rep of π1(Wp/q) is a smooth

point of a unique non-trivial curve in the character variety X(Wp/q).

We will start in section 3.1.1 by showing this for the character of each non-abelian

reducible representation ρ ∈ R(Wp/q), using work of Heusener and Porti (cf. [HP04])

by calculating the associated twisted Alexander polynomials4. Note that this a subtle

question, as it asks whether a reducible character can be deformed into irreducible

ones. We therefore obtain that each character of a non-abelian reducible p-rep indeed

contributes to the total Culler-Shalen seminorm.

3It does not contain any closed essential surfaces.

4This will in fact show that the character lies in the transverse intersection of a non-trivial with

a trivial curve.

61

After describing a different parametrization for p-reps in section 3.1.2 which lends

itself better to cohomological calculations than the one used in chapter 2, we will then

proceed in section 3.1.3 to calculate the cohomology of π1(Wp/q) with coefficients in slρ2

(the Lie algebra sl2(C) viewed as a π1(Wp/q) module via the map Ad ◦ρ) for a given

non-abelian p-rep ρ ∈ R(Wp/q).

More specifically, we will determine the dimension of Z1(Wp/q; slρ2) at a p-rep ρ

and show that it is equal to 4. Lemma 1.4.3 (for ρ irreducible) and Theorem 1.4.4 (for ρ

reducible) then imply that χρ is a simple (i.e. smooth) point of X(Wp/q). See section 1.4

for the theoretical background.

3.1.1 Smoothness of the characters of non-abelian reducible p-reps

We will show:

Proposition 3.1.1 Let ρ ∈ R(Wp/q) be a non-abelian reducible PSL2(C) representa-

tion. For p/q 6= 4, the character χρ of ρ is contained in precisely two irreducible com-

ponents of X(Wp/q), both of which are curves, one trivial, the other non-trivial. In

addition, χρ is a smooth point of both curves and the intersection at χρ is transverse.

When p/q = 4, the above is true for all non-abelian reducible representations ρ with

trace ρ(µ0) 6= ±2.

Since every non-abelian reducible SL2(C) representation ρ ∈ R(Wp/q) induces a non-

abelian reducible PSL2(C) representation ρ ∈ R(Wp/q), this then implies:

Corollary 3.1.2 Every non-abelian reducible SL2(C) character χρ ∈ X(Wp/q) lies on a

non-trivial curve X0 ⊂ X(Wp/q).

Proof. Let ρ ∈ R(Wp/q) be a non-abelian reducible representation, and let ρ ∈

R(Wp/q) be its image under the natural projection π : R(Wp/q) → R(Wp/q). Propo-

sition 3.1.1 now shows that ρ lies on a unique non-trivial curve X0 ⊂ X(Wp/q), and we

only need to show that this curve lifts to a non-trivial curve X0 ⊂ X(Wp/q). But this

is immediately clear, since ρ lifts to ρ by construction, and π is a regular covering map

onto its image (cf. [BZ98, Section 3]). �

62

The proof of Proposition 3.1.1 uses the twisted Alexander polynomial associated

to a diagonal representation with the same trace as ρ and relies on the results in [HP04],

where the authors show the following for a 3-manifold M with torus boundary which is

a rational homology circle, i.e. H1(M ; Q) = Q:

Theorem 3.1.3 ([HP04, Theorems 1.2 and 1.3]) Let ρ ∈ R(M) be a reducible repre-

sentation and assume its image is contained in {±(a 0b a

)| a, b ∈ C}. Let α : π1(M) → C∗

be the map taking each element γ to the square of the upper left entry in ρ(γ), and let

ρα be the diagonal representation obtained from ρ by setting all non-diagonal matrix

entries zero. Finally, let δ ∈ π1(M) be any element which is mapped by the Hurewicz

homomorphism to a generator of H1(M ; Z)/ tors(H1(M ; Z)).

If α(δ) is a simple zero of the twisted Alexander polynomial associated to ρα, then

1. ρα is contained in precisely two irreducible components of R(M), one of dimension

4 containing irreducible representations and another of dimension 3 containing

only abelian ones. In addition, ρα is a smooth point of both subvarieties and the

intersection at the orbit of ρα is transverse.

2. The character χραof ρα is contained in precisely two irreducible components of

X(M), both of which are curves and are the quotients of the components of R(M)

in (1). In addition, χραis a smooth point of both curves and the intersection at

χραis transverse. �

We will start by determining all possible non-abelian reducible representations of

π1(Wp/q):

Lemma 3.1.4 Let ρ be a non-abelian reducible representation in R(Wp/q). Then up to

conjugation, ρ has the form

ρ(µ0) = ±

s 0

c s−1

and ρ(µ1) = ±

u 0

d u−1

,

where either

63

1. s2 = 1, c 6= 0 and u2 = 12q (−p+ 2q ±

√p(p− 4q)) (where p/q = 0 cannot arise),

or

2. u2 = 1, d 6= 0 and sp = ±1, but s2 6= 1.

Proof. Let ρ ∈ R(Wp/q) be non-abelian reducible. Since it is reducible, we can conju-

gate it into lower triangular form as in the claim. The first group relation of π1(Wp/q)

(cf. equation (2.2)) now implies that either f := du(1− s2) + cs(u2 − 1) = 0 (in which

case ρ would be abelian, a contradiction), or one of s2 = 1 or u2 = 1 — but not both

(otherwise ρ would again be abelian).

If s2 = 1, we hence have u2 6= 1 and c 6= 0. We can now conjugate by diagonal

matrices to let c = 1. Then ρ(λ0) = ±(

1 0s−1u−2(u2−1)2 1

), and we get

±I = ρ(µ0)pρ(λ0)q = ±

sp 0

sp+1p sp

1 0

qs−1u−2(u2 − 1)2 1

= ±

sp 0

psp+1 + spqs−1u−2(u2 − 1)2 sp

.

Since s2 = 1, the lower left entry implies qu4 + u2(p − 2q) + q = 0, and therefore

u2 = 12q (−p + 2q ±

√p(p− 4q)) as claimed. Note that these roots are distinct unless

p/q = 0 or p/q = 4. When p/q = 0, this implies that u2 = 1, a contradiction since ρ is

non-abelian. When p/q = 4, we get u2 = −1 is a double root of the above polynomial,

and this gives a valid non-abelian reducible representation.

In the remaining case, when s2 6= 1, we have u2 = 1 and can assume d = 1.

Then ρ(λ0) = ±I and we immediately get ±I = ρ(µ0)pρ(λ0)q = ρ(µ0)p which implies

sp = ±1. �

Following [HP04], we now determine the twisted Alexander polynomial associated

to the diagonal representation ρα obtained from a non-abelian reducible representation

ρ ∈ R(Wp/q) as in Lemma 3.1.4 by setting all non-diagonal matrix entries to be zero.

In particular, this representation has the same character as ρ, i.e. χρα= χρ.

64

Lemma 3.1.5 Let ρ be a non-abelian reducible representation in R(Wp/q), and let ρα

be the diagonal representation with the same character as ρ as described above. The

twisted Alexander polynomial associated to ρα is given by:

∆Φσ(t) .=

t− 1 if s2 6= 1,

qt2 + (p− 2q)t+ q if s2 = 1.

Proof. We follow the notation of [HP04] to facilitate comparison for the reader. First

note that H1(Wp/q; Z) ∼= Z/p ⊕ Z = 〈µ0 〉 ⊕ 〈µ1 〉, where we use the same symbols for

the generators of π1(Wp/q) and their images under abelianization. We now define the

following homomorphisms:

α : π1(Wp/q) → H1(Wp/q; Z) → C∗, µ0 7→ s2, µ1 7→ u2,

φ : π1(Wp/q) → Z = 〈 t 〉 , µ0 7→ 1, µ1 7→ t,

σ : π1(Wp/q) → torsH1(Wp/q; Z) α→ C∗, µ0 7→ s2, µ1 7→ 1,

where s and u are the upper left entries of ρ(µ0) and ρ(µ1), respectively. Using these

maps, we can now define a map Φσ : π1(Wp/q) → C[t±]∗ to the units of the ring of

Laurent polynomials C[t±1] by Φσ(γ) := σ(γ)tφ(γ) for γ ∈ π1(Wp/q). This map can be

linearly extended to the group ring Zπ1(Wp/q) which allows us to define the twisted

Alexander polynomial ∆Φσ(t) ∈ C[t±1] (cf. [HP04, Section 2] or e.g. [Wad94] for more

background). Since we have a presentation with two generators µ0 and µ1 and two

relators

r1 = µ0µ1µ−10 µ−1

1 µ−10 µ1µ0µ1µ

−10 µ−1

1 µ0µ1µ0µ−11 µ−1

0 µ−11

and r2 = µp0λq0, the twisted Alexander polynomial can be obtained as the greatest

common divisor of the entries of the twisted Fox Jacobian JΦσ :

JΦσ =

∂r1∂µ0

∂r1∂µ1

∂r2∂µ0

∂r2∂µ1

Φσ

=

s−2t−1(t− 1)2(s2 − 1) s−2t−1(1− t)(s2 − 1)2∑p−1k=0 s

2k + s2pqt−1(t− 1)2 s2pqt−1(1− t)(s2 − 1)

.

65

This matrix can be obtained by hand by first determining the Fox derivatives of the

group relators (cf. [Fox53] or [BZ03, Ch. 9B] for background on Fox free differential

calculus), and then applying the map Φσ to the obtained elements of the group ring

Zπ1(Wp/q).

We now have two cases: If s2 = 1, we get

JΦσ =

0 0

p+ qt−1(t− 1)2 0

,

and thus that ∆Φσ(t) = p+qt−1(t−1)2 .= qt2+(p−2q)t+q (where .= denotes equivalence

up to multiplication by units of C[t±1]). Note that this is in fact the untwisted Alexander

polynomial of Wp/q, as s2 = 1 implies that σ is the trivial 1-dimensional representation

(cf. also e.g. [HMW92], taking into account that the authors work with the left-handed

Whitehead link).

If on the other hand s2 6= 1, we have s2p = 1 and∑p−1

k=0 s2k = 0 which yields

JΦσ =

s−2t−1(t− 1)2(s2 − 1) s−2t−1(1− t)(s2 − 1)2

qt−1(t− 1)2 qt−1(1− t)(s2 − 1)

.

As a result, ∆Φσ(t) = t−1(t− 1) .= t− 1 in this case. �

We can now proceed to prove the main result:

Proof (of Proposition 3.1.1). Let ρ ∈ R(Wp/q) be non-abelian reducible. Lemma 3.1.4

shows that s2 6= 1 implies u2 = 1 (i.e. u2 is the root of the polynomial t−1), and s2 = 1

implies u2 is a root of qt2 + (p− 2q)t+ q. But these are precisely the twisted Alexander

polynomials found in Lemma 3.1.5 for these cases, and so we have that u2 is a simple

root of the twisted Alexander polynomial, unless s2 = 1 and p/q = 4 (in which case

u2 = −1 is a double root). Apart from this exception, Theorem 3.1.3 with δ = µ1

then shows that the character of the diagonal representation ρα (which is equal to the

character of ρ) satisfies the claim of the Theorem. �

66

3.1.2 A partially diagonal parametrization of p-reps

We now determine a parametrization of p-reps which lends itself well to the co-

homology calculations of the next section.

Recall that

π1(Wp/q) =⟨µ0, µ1 | µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0µ1 = µ1µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0,

µp0λq0 = 1

⟩,

(3.3)

where λ0 = µ1µ0µ−11 µ−1

0 µ−11 µ0µ1µ

−10 .

Let ρ ∈ R(Wp/q) ⊂ R(W ) be a p-rep (recall that ρ is non-abelian). We will

identify it with the partial p-rep ρ′ ∈ R(W ) satisfying ρ′(µp0λq0) = I, which we will

also denote by ρ. We have seen in chapter 2 that ρ(µ0) 6= ±I can be assumed to be

non-parabolic, and so we can conjugate ρ into the subvariety D(Wp/q) ⊂ R(Wp/q) (cf.

section 1.5.2.2). In other words, we can assume up to conjugation that

ρ(µ0) =

s 0

0 s−1

and ρ(µ1) =

a ab− 1

1 b

, (3.4)

where s ∈ C∗, a, b ∈ C, subject to the condition (obtained from the first group relation)

r1 = s4a2b− s4a− a2bs2 + s2ab2 + b− ab2 = 0. (3.5)

Furthermore, we have ρ(λ0) =(t 00 t−1

)with t ≡r1 a/b. See section 1.5.2.2 for the details.

The fact that ρ is diagonal on the peripheral subgroup of the first boundary torus will

be of great importance in section 3.1.3.

Note that if ρ is irreducible, we have ab 6= 1. The case ab = 1 corresponds to

non-abelian reducible p-reps, which lie in the intersection C(Wp/q) ∩D(Wp/q).

We now make use of the fact that ρ (as a p-rep) sends µ1 to a parabolic element,

i.e. trace ρ(µ1) = ±2. It therefore lies in one of the following subvarieties of D(Wp/q):

D+ := D(Wp/q) ∩ {b = 2− a}, (3.6)

D− := D(Wp/q) ∩ {b = −2− a}. (3.7)

67

We will study both cases at once, respecting the order of the signs ± versus ∓ through-

out.

For ρ ∈ D±, we get

ρ(µ0) =

s 0

0 s−1

and ρ(µ1) =

a −(a∓ 1)2

1 ±2− a

(3.8)

The condition r1 = 0 then simplifies to:

r±1 = (a− 1)[−(s2 − 1)2a2 ± (s2 − 1)(s2 − 3)a− 2

]= 0. (3.9)

Note that this implies a 6= 0 and a 6= ±2 (respecting the sign, we would otherwise have

r±1 = 2s4 6= 0). We now need to distinguish between the reducible and the irreducible

case:

Note first that ρ ∈ D±(Wp/q) is reducible iff a = ±1. It is then easy to see that in

this case the polynomial r1 is identically zero, and poses therefore no further condition

on s and a. Then ρ ∈ C(Wp/q)∩D(Wp/q), and a = u = ±1 is equal to the eigenvalue of

ρ(µ1). The condition r2 = 0 then implies sp = 1, just as we found in chapter 2.

If ρ ∈ D± is irreducible, we have a 6= ±1, and we can further simplify r±1 to:

r±1 = −(s2 − 1)2a2 ± (s2 − 1)(s2 − 3)a− 2. (3.10)

For simplicity, we will denote this polynomial also by r±1 , since there will be no danger

of confusion.

It now remains to use the fact that ρ(µp0λq0) = I. Using the diagonal form and

t ≡r1 a/b this implies5

r±2 ≡r±1 sp(

a±2−a

)q− 1 = 0. (3.11)

But note that we have already solved the system r1 = r2 = 0 in chapter 2 in terms of

s and t (where the corresponding polynomials were named k1 and k2). In the following

5Note that t 6= 0 implies a 6= 0 and a 6= ±2 (the sign corresponding to ρ ∈ D±).

68

we will use the condition r±1 = 0 to reduce all appearing polynomials in a and s to their

unique normal forms with respect to division by r±1 in C[a, s] (see e.g. [CLO97]), since

this can be done explicitly with the help of computer algebra software6. In section 3.2,

we will then use the characterization of the eigenvalue s by the condition resp,q(s) = 0

in the irreducible, and the condition sp = 1 in the reducible case.

3.1.3 The twisted cohomology at p-reps

We now determine a presentation for the cohomology group H1 := H1(Wp/q; slρ2)

(cf. section 1.4).

First we determine the coboundaries B1 := B1(Wp/q; slρ2). Let

e1 =

0 1

0 0

, e2 =

1 0

0 −1

and e3 =

0 0

1 0

(3.12)

be a basis for the Lie algebra sl2 := sl2(C). For a matrix A ∈ sl2, let uA ∈ B1 be the

1-coboundary defined by uA(γ) = A − γ · A for γ ∈ π1(Wp/q). Given a representation

ρ ∈ D, the coboundaries are then generated by

ue1 : µ0 7→

0 1− s2

0 0

, µ1 7→

a 1− a2

1 −a

, (3.13)

ue2 : µ0 7→

0 0

0 0

, µ1 7→

2(1− ab) −2a(1− ab)

−2b −2(1− ab)

, (3.14)

ue3 : µ0 7→

0 0

1− s−2 0

, µ1 7→

b(1− ab) (1− ab)2

1− b2 −b(1− ab)

. (3.15)

Since cochains are determined by their images on the generators, we will implicitly

use the isomorphism τ : sl22 → C6 given byx1 x2

x3 −x1

,

y1 y2

y3 −y1

7→ (x1, x2, x3, y1, y2, y3) (3.16)

6We have mainly used the program Maple in its current version 10 (and previous versions,

verifying our results).

69

to identify a cochain u with an element of C6. We can therefore think of Z1 :=

Z1(Wp/q; slρ2) and B1 as subspaces of C6.

All coboundaries are now of the form α1ue1 + α2ue2 + α3ue3 with αi ∈ C. Seen

as elements of C6, they hence lie in the span of the row vectors of the matrix

B :=

0 1− s2 0 a 1− a2 1

0 0 0 2(1− ab) −2a(1− ab) −2b

0 0 1− s−2 b(1− ab) (1− ab)2 1− b2

. (3.17)

Since s 6= ±1, one then sees that the rank of this matrix, and hence the dimension of

the subspace of coboundaries, is 3. Note that this remains true when ρ is reducible, as

then ab = 1 implies b 6= 0 and B becomes

Bred =

0 1− s2 0 a 1− a2 1

0 0 0 0 0 −2b

0 0 1− s−2 0 0 1− b2

. (3.18)

Let (B1)⊥ be the orthogonal complement of the subspace of coboundaries B1 in

C6 with respect to the standard inner product. For each cohomology element u ∈ H1

there then is a unique representative u ∈ (B1)⊥ such that u = u+B1. The orthogonal

complement is given by the kernel of the matrix B, and we can therefore identify H1

with a subspace of (B1)⊥ in C6. In particular, we can obtain it as the kernel of the

linear map given by the matrix B adjoined with additional rows corresponding to the

cochain versions of the group relations.

We therefore examine the effect of the first group relation on cochains: Using

w1 := µ1µ0µ1µ−10 µ−1

1 µ−10 µ1µ0 = µ0µ1µ

−10 µ−1

1 µ−10 µ1µ0µ1 =: w2, we see that each cochain

u has to satisfy the matrix equation u(w1)− u(w2) = 0.

To simplify the exposition, we will from now on only consider the case ρ ∈ D+, i.e.

trace ρ(µ1) = +2 (b := 2−a). The negative trace case can be dealt with in a completely

analogous way.

The upper left entry of u(w1) − u(w2) yields a polynomial in x1, x2, x3, y1, y2

and y3 with coefficients in C[a±1, s±1] which have to vanish. When ρ is irreducible (and

70

hence a 6= 1), we can divide by the common nonzero factor of s−8(s2 − 1)(a − 1) and

reduce all coefficients modulo r+1 to obtain the following equation (written in vector

form):

−4(a3s2 − a3 − 7a2s2 + 7a2 + 2as4 + 8as2 − 14a+ 4s2 + 8)

−2(a2s2 − a2 − 2as4 − as2 + 3a+ s4 − 2s2 − 2)

2(a4s2 − a4 − 9a3s2 + 9a3 + 24a2s2 − 29a2 − as6 − 16as2 + 39a+ s6 − 2s4 − 8s2 − 18)

2s2(−6as2 + 3a2s2 − as4 − a2 + 3a− 2)

as6 + 2as4 − 2as2 + a− 2s4 + 4s2 − 2

−3a3s2 + a3 + 16a2s2 − 8a2 − 4as4 − 19as2 + 17a− 2s2 − 10

T

x1

x2

x3

y1

y2

y3

= 0.

(3.19)

Denote the first coefficient (corresponding to x1) by ψ1. We will verify that ψ1 does not

vanish at a representation ρ ∈ D+: Calculating7 a reduced Groebner basis with respect

to the pure lexicographic order a > s for the ideal generated by ψ1 and r+1 , we find that

its first generator is a power of s, implying that the only common zeros with r+1 lie in

the subspace defined by s = 0, a contradiction to our assumptions.

Thus, we see that the vector in (3.19) is linearly independent from the row vectors

in the matrix B. Also note that the other entries of u(w1)−u(w2) yield equation which

can be shown to be consequences of the coboundary conditions and equation (3.19).

If, on the other hand, ρ is reducible (but non-abelian), we find that only the lower

left entry of u(w1) = u(w2) gives a non-vanishing equation, which we write in vector

form as

0

−2(s2 − 1)

0

−2(s2 − 1)2

s−2(s2 − 1)2(s4 − s2 − 1)

0

T

x1

x2

x3

y1

y2

y3

= 0. (3.20)

This vector can now easily be seen to be linearly independent from the row vectors in

the matrix Bred in (3.18) (recall a = b = 1 in this case).

7With the help of Maple

71

Note that the above can be interpreted as saying that the dimension of the first

cohomology group H1(W ; slρ2) of the unfilled Whitehead-link complement is 2 at non-

abelian representations ρ ∈ D+: In the irreducible case, the 4× 6-matrix formed by B

and the row corresponding to equation (3.19) is a presentation matrix for H1(W ; slρ2)

as a subspace of C6. In the reducible case, a presentation matrix for H1(W ; slρ2) is

given by the 4 × 6-matrix formed by Bred from (3.18) (for a = b = 1) together with

the row corresponding to equation (3.20). Using Lemma 1.4.3, we therefore obtain the

intermediate result:

Proposition 3.1.6 The character χρ of a representation ρ ∈ R(W ) satisfying χρ(µ0) 6=

±2 and χρ(µ1) = ±2 is a smooth point of X(W ). Furthermore, χρ is contained in a

unique algebraic component of dimension 2.

We now study the condition forced on the cochains after applying p/q-filling on the

first cusp. Letting u(µ0) = ( x1 x2x3 −x1

) and u(µ1) =( y1 y2y3 −y1

)as before, we first determine

the image u(λ0) = u(µ1µ0µ−11 µ−1

0 µ−11 µ0µ1µ

−10 ). A direct calculation shows that

u(λ0) =:

l1 l2

l3 −l1

, (3.21)

where the normalized entries l1, l2 and l3 are given by

l1 =− 4s−6(a3s2 − a3 − 2a2s2 + 4a2 − as6 − 2as4 − as2 − 5a+ s6 + 2s4 + 2s2 + 2)

− 2s−8(a2s2 − a2 − as2 + 3a− 2s6 − 2s4 − 2s2 − 2)

+ 2s−4(a4s2 − a4 − 3a3s2 + 5a3 − 3a2s2 − 7a2 + 2as4 + 11as2 + a− 2s4 − 2s2 + 2)

− 4s−2(a3s2 − a3 − 3a2s2 + 4a2 + 2as2 − 5a+ 2)

− s−4(2a2s2 − 2a2 − as6 − 3as2 + 6a− 4s4 − 2s2 − 4)

+ 2s−6(a4s2 − a4 − 6a3s2 + 7a3 + 12a2s2 − 19a2 − 5as2 + 23a− 2s4 − 6s2 − 10),

(3.22)

72

l2 =4s−6(a4s2 − a4 − a3s2 + 3a3 − 11a2s2 + 3a2 + 17as2 − 15a+ 2s2 + 10)

+ s−8(2a3s2 − 2a3 + 4a2 − 15as4 + a− s4 − 8s2 − 3)

− s−4(2a5s2 − 2a5 − 4a4s2 + 8a4 − 12a3s2 − 6a3 + 32a2s2 − 12a2 − as4 − 20as2 + 21a+ s4 − 9)

+ 2s−2(2a3s2 − 2a3 − 5a2s2 + 7a2 + 2as2 − 8a+ s2 + 3)a

+ s−4(2a3s2 − 2a3 − 5a2s2 + 7a2 − 5as4 + 2as2 − 7a+ s4 − s2 + 2)

− s−6(2a5s2 − 2a5 − 11a4s2 + 13a4 + 18a3s2 − 32a3 + 4a2s2 + 30a2 − 28as2 + 2a− s2 − 11)

(3.23)

and

l3 =− 4s−6(a2s2 − a2 − 2as8 + 2as6 − as2 + 3a+ 2s8 − 6s6 − 2s4 − 2s2 − 2)

+ s−2(as6 − 6as4 + 7as2 − 2a− s6 + 8s4 − 15s2 + 4)

+s−4(2a3s2 − 2a3 − 8a2s2 + 12a2 + as12 − 6as10 + 11as8 − 4as6 − 4as4

+2as2 − 22a− s12 + 8s10 − 19s8 + 12s6 + 12s4 + 12s2 + 12)

− 2s−2(2a2s2 − 2a2 − 2as4 − 3as2 + 7a+ 2s4 − 2s2 − 6)

+ s−4(as10 − 7as8 + 13as6 − 12as4 + 6as2 − a− s10 + 9s8 − 22s6 + 22s4 − 12s2 + 2)

+ s−6(2a3s2 − 2a3 − 13a2s2 + 15a2 + as8 + 5as6 + 11as2 − 31a− s8 − 3s6 + 10s4 + 14s2 + 18).

(3.24)

The relation µp0λq0 = 1 then translates8 to u(µp0λ

q0) = ( 0 0

0 0 ), where

u(µp0λq0) = u(µp0) + µp0 · u(λ

q0) =

=(1 + µ0 + · · ·+ µp−1

0

)· u(µ0) + µp0

(1 + λ0 + · · ·+ λq−1

0

)· u(λ0)

=

px1 + ql1 x2∑p−1

k=0 s2k + s2pl2

∑q−1j=0 t

2j

x3∑p−1

k=0 s−2k + s−2pl3

∑q−1j=0 t

−2j −px1 − ql1

.

(3.25)

Now s 6= ±1 implies that we can usep−1∑k=0

s2k =s2p − 1s2 − 1

(3.26)

andp−1∑k=0

s−2k = s−2(p−1)p−1∑k=0

s2k = s−2(p−1) s2p − 1s2 − 1

. (3.27)

8This computation is the reason for our choice of a diagonal form for ρ(µ0) and ρ(λ0), as we

could otherwise not have arrived at a general form in terms of p and q.

73

We now need to distinguish between the reducible and irreducible case: When ρ

is reducible, a = 1 implies t = 1 and hence∑q−1

j=0 t2j = q. But then sptq = 1 also implies

sp = 1, and equation (3.25) becomes0 0

0 0

=

px1 + ql1 ql2

ql3 −px1 − ql1

. (3.28)

But note that a = 1 implies that l1 = (s2 − s−2)y2, l2 = 0 and l3 = −2s−2x2 + 2(s−2 −

1)y1 + (s−4 − 2 + s2)y2. It is then easy to see that ql3 = 0 is already a consequence of

the coboundary conditions (3.18), and so the only new condition remaining from (3.28)

is (p/q)x1 + l1 = 0. Adjoining this condition in row vector form together with the

vector corresponding to equation (3.20) to the matrix Bred from (3.18), we then obtain

a presentation matrix for the cohomology group H1(Wp/q; slρ2), viewed as a subspace

of C6:0 1− s2 0 1 0 1

0 0 0 0 0 −2

0 0 1− s2 0 0 0

0 2(1− s2) 0 2s−2(−s4 + 2s2 − 1) s−2(s4 − s2 − 1)(s2 − 1)2 0

p/q 0 0 0 s−2(s4 − 1) 0

. (3.29)

The rank of this matrix can easily be seen to be 5 for all s 6= ±1, and we therefore

obtain

dimCH1(Wp/q; sl

ρ2) = 1. (3.30)

On the other hand, if ρ is irreducible, we can use t 6= ±1 to rewrite equa-

tion (3.25) as0 0

0 0

=

px1 + ql1s2p−1s2−1

x2 + s2p t2q−1t2−1

l2

s−2(p−1) s2p−1s2−1

x3 + s−2pt−2(q−1) t2q−1t2−1

l3 −px1 − ql1

=

px1 + ql1s2p−1s2−1

x2 + 1−s2p

t2−1l2

s2 1−t2q

s2−1x3 − t2 1−t2q

t2−1l3 −px1 − ql1

=

px1 + ql1 (s2p − 1)(

1s2−1

x2 − 1t2−1

l2

)s−2p(s2p − 1)

(s2

s2−1x3 − t2

t2−1l3

)−px1 − ql1

.

(3.31)

74

From the upper right and lower left entries of (3.25) we now see that either s2p = 1

(which we know cannot be true by Lemma 2.6.6, since s is also a root of resp,q), or

(t2 − 1)x2 = (s2 − 1)l2 and s2(t2 − 1)x3 = t2(s2 − 1)l3. After some careful algebraic

manipulations, it can be verified that these last two conditions are in fact consequences

of the coboundary conditions (3.17) and equation (3.19). The only new information

comes therefore from the upper left matrix entry. Since q > 0, we can rewrite this in

the form l1 + (p/q)x1 = 0, which gives us (after normalizing with respect to r+1 ) the

condition

p/q − 4s−6(a3s2 − a3 − 2a2s2 + 4a2 − as6 − 2as4 − as2 − 5a+ s6 + 2s4 + 2s2 + 2)

−2s−8(a2s2 − a2 − as2 + 3a− 2s6 − 2s4 − 2s2 − 2)

2s−4(a4s2 − a4 − 3a3s2 + 5a3 − 3a2s2 − 7a2 + 2as4 + 11as2 + a− 2s4 − 2s2 + 2)

−4s−2(a3s2 − a3 − 3a2s2 + 4a2 + 2as2 − 5a+ 2)

−s−4(2a2s2 − 2a2 − as6 − 3as2 + 6a− 4s4 − 2s2 − 4)

2s−6(a4s2 − a4 − 6a3s2 + 7a3 + 12a2s2 − 19a2 − 5as2 + 23a− 2s4 − 6s2 − 10)

T

x1

x2

x3

y1

y2

y3

= 0.

(3.32)

Adjoining the two row vectors corresponding to equations (3.19) and (3.32) to

the matrix B, we now again obtain a presentation matrix for H1(Wp/q; slρ2). With some

care9, we can row-reduce this matrix to the form

ψ1 0 0 0 ∗ ∗

0 s2 − 1 0 0 −1 −(a− 1)−2

0 0 s−2 − 1 0 s2(a− 1)2 s2

0 0 0 (a− 1)2 −a(a− 1)2 a− 2

0 0 0 0 ∗ ∗

, (3.33)

where each ∗ denotes a non-zero polynomial in a and s (normalized modulo r+1 ), and

ψ1 is the first coefficient of the vector in equation (3.19). This shows that the rank of

this matrix is 5, and we therefore find that equation 3.30 also holds in the irreducible

case.

Since we therefore have dimZ1(Wp/q; slρ2) = 4 both in the irreducible and reducible

case, Lemma 1.4.3 and Theorem 1.4.4 now show:

9Making sure no polynomial used in division can be zero on D+.

75

Proposition 3.1.7 The character χρ of a p-prep ρ ∈ R(Wp/q) is a smooth point

of Xirr(W ) and is contained in a unique non-trivial curve. Furthermore, we have

TZarχρ

(X(Wp/q)) ∼= H1(Wp/q; slρ2). �

We omit the proof for the negative trace case ρ ∈ D− as it proceeds analogously.

Besides the smoothness, the presentation matrices obtained above give us an

explicit parametrization for non-trivial cocycles in Z1(M ; slρ2) at a given representation

ρ ∈ D+. This will be needed in the following section, when we will study deformations

of these representations.

3.2 Deformations of representations and degree of the trace function

Recall that the ultimate goal of this chapter is to determine the degree of the map

fµ1 = I2µ1− 4, where Iµ1(χρ) = trace ρ(µ1) is the trace function of µ1. It remains to

show that each character of a p-rep ρ ∈ D± is a zero of order 1 of fµ1 . To do this, we

will use a Taylor expansion of fµ1 using the identification of the Zariski tangent space

TZarχρ

(X(Wp/q)) with the cohomology group H1(Wp/q; slρ2) (cf. Proposition 3.1.7).

Since we have shown in the last section that Xirr(Wp/q) is smooth and positive-

dimensional at the character of a p-rep ρ ∈ D±, we can use the existence of non-trivial

cocycles to find a smooth deformation ρν of ρ0 = ρ, which admits an expansion of the

form

ρν = exp(νu1 + ν2u2 + ν3u3 + . . .

)ρ (3.34)

where uj : π1(G) → sl2(C), and u1 ∈ Z1(G; slρ2) (see e.g. [HPSP01, §3] or [Gol84]).

Developing fχρν= (trace ρν(γ))

2 − 4 using the expansion

exp(A) =∞∑j=1

Aj/j! (3.35)

for A ∈ sl2(C), we find that the trace function fµ1(χρν ) has the following Taylor expan-

sion at ρ0 (see e.g. Lemma 1.3 in [BAB01]):

fµ1(χρν ) = fµ1(χρ0) + 2Iµ1(χρ0) trace(u1(µ1)ρ0(µ1)

)ν +O(ν2). (3.36)

76

Note that in our case G := π1(Wp/q) and Iµ1(χρ0) = (trace ρ0(µ1))2 = 4 for ρ0 ∈ D±.

Since fµ1(χρ0) = 0 at a p-rep ρ0, the order of this zero will be equal to one iff there is

a u1 ∈ Z1(Wp/q; slρ2) satisfying

trace(u1(µ1)ρ0(µ1)

)6= 0. (3.37)

To prove this, note that for ρ0 ∈ D± and an arbitrary cochain u1 we have

trace(u1(µ1)ρ0(µ1)

)= trace

y1 y2

y3 −y1

a −(a∓ 1)2

1 ±2− a

= 2(a∓ 1)y1 + y2 − (a∓ 1)2y3

(3.38)

Let P be the matrix obtained by adjoining the row vector corresponding to this

expression to the presentation matrix for H1(Wp/q; slρ2). The existence of a non-trivial

cocycle u1 such that (3.38) is equal to a non-zero value c ∈ C∗ is now equivalent to

finding a solution u1 := (x1, x2, x3, y1, y2, y3) to the linear equation system

P (x1, x2, x3, y1, y2, y3)T = (0, 0, 0, 0, 0, c)T . (3.39)

and hence a non-trivial element in H1(Wp/q) represented by a cocycle u1 which satisfies

(3.37).

In particular, if we find that detP 6= 0, the system (3.39) will have a unique

solution u1 for any given c ∈ C∗.

Again restricting attention to ρ0 ∈ D+, we will first discuss the case when ρ0 is

reducible: In this case, the matrix P is given by

P =

0 1− s2 0 1 0 1

0 0 0 0 0 −2

0 0 1− s2 0 0 0

0 2(1− s2) 0 2s−2(−s4 + 2s2 − 1) s−2(s4 − s2 − 1)(s2 − 1)2 0

p/q 0 0 0 s−2(s4 − 1) 0

0 0 0 0 1 0

, (3.40)

and we find detP = (4p/q)s−4(s2 − 1)2(s4 − 2s2 + 2). Now recall that the eigenvalue s

of a reducible p-rep must satisfy sp = 1, s 6= ±1. None of the roots of s4−2s2 +2 = 0 lie

77

on the unit circle, and so detP 6= 0 unless p = 0. When p = 0, the trace in (3.38) will

indeed be zero for all cocycles u1, and hence the order of a zero of fµ1 at the character

of a reducible non-abelian p-rep is indeed greater than 1. This also reflects the fact that

there is a whole curve of non-abelian reducible p-reps for s ∈ C∗ − {±1} in this case

(since p = 0, q = 1 implies sptq = t = 1, and there are no further conditions on s). We

summarize our result:

Proposition 3.2.1 When p 6= 0, each character of a non-abelian reducible p-rep ρ0 ∈

R(Wp/q) is a zero of order 1 of fµ1 |Xirr(Wp/q). �

We will now discuss the remaining case, when ρ0 ∈ D+ is irreducible. Appending

the row vector corresponding to the expression (3.38) to the matrix (3.33) and applying

Gauss-Jordan to reduce the matrix, we obtain the 6× 6-matrix

P =

ψ1 ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗

2a2 − 2a+ 1 (1− a)−1(a3 − 3a2 + 5a− 5)

(3.41)

Denote by d the determinant of the lower right 2× 2-block of this matrix. We wish to

show that this determinant is not zero for ρ0 ∈ D+. To show this, we used the computer

algebra system Maple to calculate a Groebner basis of the ideal generated by r+1 and

the numerator of d with respect to a lexicographic ordering with a > s. We find10 that

the first basis element is given by s30d1d2 where11

d1 = p(p− 4q)s4 + (−6p2 + 24pq − 32q2)s2 + p(p− 4q) (3.42)

10We remark that this computation took more than 24 hours on a modern PC.

11It is interesting to note that a version of d1 for q = 1 already appeared in section 2.6.3 as a

factor in equation (2.52).

78

and

d2 = 22s40 − 438s38 + 4185s36 − 24868s34 + 101875s32 − 304088s30

+ 683740s28 − 1182928s26 + 1598312s24 − 1708564s22 + 1466502s20

− 1027864s18 + 595850s16 − 286072s14 + 115452s12 − 41808s10

+ 14586s8 − 4582s6 + 1097s4 − 164s2 + 11. (3.43)

If d were to vanish at a representation ρ0 ∈ D+, each of the polynomials in the Groebner

basis, and in particular the first element, would have to vanish. We will therefore show

that neither d1 nor d2 vanish at a representation ρ0 ∈ D+, which shows that d cannot

vanish there. For this, we will make use of the fact that the parametrization of any

representation ρ0 ∈ D± also satisfies resp,q = 0 (as determined in section 2.6). It suffices

therefore to show that resp,q has no roots in common with d1 and d2.

As for d1, we note that its four roots are

±

((3(p− 2q)2 + 4q2

)±

(2|p− 2q|

√2(p− 2q)2 + 8q2

))1/2

√p(p− 4q)

(3.44)

It is easy to see that in fact these roots are all real-valued when p > 4q > 0 or when

p < 0 < 4q, and all imaginary when 0 < p < 4q. To see this, simply note that

(3(p− 2q)2 + 4q2

)2 −(2|p− 2q|

√2(p− 2q)2 + 8q2

)2= p2(p− 4q)2 ≥ 0 (3.45)

which shows that the numerator in (3.44) is always real-valued.

But note that for p > 4q > 0 or p < 0 < 4q, apart from ±1, none of the non-zero

roots of resp,q are real-valued. Similarly for 0 < p < 4q, none of its non-zero roots are

pure imaginary. So d1 in fact does not share any roots with resp,q.

It now remains to be shown that the polynomial d2 has no common roots with

resp,q. To see this, first observe that d2 is an irreducible polynomial, while its leading

coefficient is 22 6= 1. On the other hand, while resp,q is in general not irreducible12, it

12For example when q = 1 and p is even.

79

is a monic polynomial with integer coefficients — and therefore so are its irreducible

factors. We conclude that all roots of resp,q are algebraic integers, while this is not true

for any of the roots of d2. This shows that d2 and resp,q have no roots in common. We

show the 40 roots of d2 in figure 3.1.

Figure 3.1 The roots of the polynomial d2

We can therefore conclude that the 2×2 minor d is indeed non-zero at characters

of p-reps, and that hence rankP = 6 and detP 6= 0, giving us:

Proposition 3.2.2 Each character of an irreducible p-rep ρ0 ∈ R(Wp/q) is a zero of

order 1 of fµ1 |Xirr(Wp/q). �

This then completes the proof of Theorem 3.0.11, showing that the minimal semi-

norm s is given by the number of characters of p-reps.

Chapter 4

The total Culler-Shalen seminorm

The goal of this chapter is to completely determine the total Culler-Shalen semi-

norm of each manifold Wp/q = Wrh(p/q, · ) for p odd.

Using Lemma 1.2.7, we have for the Culler-Shalen seminorm associated to a non-

trivial curve Xi ⊂ X(Wp/q)

‖γ‖Xi=

∑j

aij∆(γ, βj), (4.1)

where the sum is over the (finite) set of boundary slopes βj of the manifold Wp/q and the

aij are non-negative even integers independent of γ ∈ π1(Wp/q). Considering the sum

over all non-trivial curves in the character variety, we obtain for the total seminorm:

‖γ‖ =∑j

aj∆(γ, βj), (4.2)

with non-negative even integers aj . To determine the total seminorm, it therefore suffices

to determine the boundary slopes βj and the corresponding coefficients aj . To accom-

plish this, we will use the fact that the seminorm of certain slopes can be explicitly

determined relative to the minimal total seminorm (which we obtained in chapter 3).

Setting these values equal to the corresponding expressions from equation (4.2), we

will obtain a system of linear equations which then can be solved for aj and s. This

technique goes back to Mattman, who successfully used it to determine the character

varieties Culler-Shalen seminorms of (−2, 3, n) pretzel knot exteriors (cf. [Mat02]).

81

In our situation, it will turn out that the knowledge of the upper bound for s (cf.

table 3.1 in chapter 3) is often already sufficient to completely solve the linear system.

We then obtain that this upper bound for s is in fact attained, which then implies that

Conjecture 2.6.8 holds when p is odd.

The organization of this chapter is as follows: In section 4.1, we determine the

set of boundary slopes of Wp/q. In section 4.2, we then use special knowledge about

certain Seifert fillings of Wp/q to determine their corresponding seminorms. This will

be obtained through an explicit count of the characters of the underlying base orbifold

fundamental groups. The knowledge of these specific values for the seminorm in turn

enables us in section 4.3 to determine the coefficients aj from solving a system of linear

equations. The final results can be found in section 4.3.5, and in particular in table 4.14

on page 111.

4.1 Boundary slopes

In this section, we will determine the set of boundary slopes of each manifoldWp/q.

To do this, we will first use the classification of normal surfaces in the Whitehead link

exterior from [Til02] to determine its set of boundary slopes. We then will determine

which of these surfaces survive to give essential surfaces in the once-filled manifolds

Wp/q.

4.1.1 Boundary slopes of the unfilled Whitehead link exterior

Since the Whitehead link is a 2-bridge link (it is L3/8 in the standard 2-bridge link

notation), its space of incompressible surfaces is well-known through the work of Floyd

and Hatcher, cf. [FH88]. Even though the authors explicitly describe all incompressible

surfaces in the exterior of a 2-bridge link, they do not compute the boundary slopes of

these surfaces, but rather remark that this should be possible in principle. In his Ph.D.

thesis [Las93], Lash showed how this can be done starting with the construction of

Floyd and Hatcher. Unfortunately, these results were never published and no tables of

boundary slopes of links have been available. After the work of this section was already

82

completed and independently of our work, the preprint [HS05] by Hoste and Shanahan

was made available, in which the authors implement (and improve) Lash’s algorithm

and include (amongst other data) tables of the boundary slopes of all 2-bridge links

with 10 or less crossings, including the (left-handed) Whitehead link L3/8 — confirming

our results.

We use the classification of embedded normal surfaces in Wrh found in [Til02]. We

refer the reader to [Thu79], the documentation to [Wee] or [Til04] for an introduction

into normal surface (Q-)theory. We shall only give a very brief overview: By choosing

an ideal topological triangulation of an orientable, non-compact1 topologically finite 3-

manifold M , one can isotope a properly embedded surface S to be transverse to each

tetrahedron of the triangulation. After further isotopies, one can often make sure2 that

the intersection of S with each tetrahedron is a collection of triangles (“T-discs”) and

quadrilaterals (“Q-discs”). A surface of this form is called normal. There are three

different kinds of Q-discs in each tetrahedron, and there can be only one type of Q-

disc in the intersection of S with a given tetrahedron. Once the type and number

of Q-discs in each tetrahedron is known, the number of T-discs can be reconstructed,

although this number may be infinite. The surface can hence be encoded by a vector

N(S) = (x(1)1 , x

(2)1 , x

(3)1 , x

(1)2 , . . . , x

(3)n ) ∈ Z3n

+ , where n is the number of tetrahedra3.

Conversely, given an admissible vector satisfying certain Q-matching equations (given

by the triangulation of the manifold), a properly embedded surface can be reconstructed.

By passing to a projective admissible solution space P (M) ⊂ R3n, the correspondence

can be made unique:

Theorem 4.1.1 (Weeks, cf. also [Til02]) Let M be an orientable non-

compact 3-manifold with a topological ideal triangulation. For each admissible integer

1Alternatively, one can consider a triangulation of an orientable compact manifold with torus

boundary by truncated tetrahedra.

2This is not always possible for non-compact surfaces.

3Such a vector is called admissible: for each i ∈ {1, . . . , n} and each j ∈ {0, 1, 2}, x(j)i ≥ 0 and

at most one of x(1)i , x

(2)i and x

(3)i is non-zero.

83

solution N ∈ P (M) there exists a unique properly embedded normal surface S (possibly

non-compact) in M with no boundary parallel components such that N(S) = N . �

The projective solution space P (M) is a convex polytope, and its vertices are called

vertex solutions.

In [Til02], the author gives an explicit description of normal surfaces in Wrh by

giving the complete list of 20 minimal representatives of the vertex solutions, in terms

of their number of Q-discs and their boundary curve information (cf. [Til02, §5.2]).

From this data, the set of all normal surfaces (and hence the set of all boundary slopes)

can be reconstructed from linear combinations between the vertex solutions. In fact,

because of symmetries4 it suffices to consider a subset of vertex solutions: We choose as

representatives the vertices 2, 3, 8, 9, 13, 16, 17 and 19, and will consider all non-trivial

integer linear combinations corresponding to the edges in the following diagram5:

/.-,()*+3 I /.-,()*+2

/.-,()*+9 II

III

/.-,()*+8IV

7654012317

V

7654012316

VIVII

BBBB

BBBB

B

7654012314 7654012313 7654012319

(4.3)

We list in table 4.1 the boundary curve information for minimal representatives for these

vertex solutions, directly taken from [Til02, Table 5.6]. For each vertex solution, we

give the topological type6, the number of boundary curves on the first/second boundary

4The dihedral group D4 acts on the space of all normal surfaces, and in particular the vertex

solutions.

5Note that the vertex solution 19 could technically be omitted, but its inclusion implies that we

already get all possible boundary slopes without having to interchange the two cusps.

6Here T1 denotes a once-punctured torus, S3 a thrice-punctured sphere and R2 a twice-punctured

RP2.

84

torus of W , and finally the slopes of the boundary curves with respect to the peripheral

systems (µ0, λ0) and (µ1, λ1) on the first and second torus boundary, respectively7.

Vertex type ∂-curves p0 q0 p1 q1

2 T1 1/0 0 −1 0 0

3 T1 0/1 0 0 0 1

8 S3 1/2 0 −1 −2 0

9 S3 2/1 2 0 0 1

13 R2 1/1 −4 −1 −2 −1

14 R2 1/1 4 1 −2 −1

16 R2 1/1 −2 −1 −4 −1

17 R2 1/1 4 1 2 1

19 R2 1/1 2 1 −4 −1

Table 4.1 Boundary curve data for minimal representatives of some vertex solutions

(from [Til02, Table 5.6])

All other normal surfaces can now be obtained from non-trivial integer linear

combinations of the form aA+ bB, where a, b ≥ 0 (but (a, b) 6= (0, 0)) and A and B are

vertex solutions. Lash shows in [Las93] that the combinations coming from the edges

in diagram (4.3) already give rise to all boundary curves of incompressible surfaces up

to projectivization. We list the corresponding combinations in table 4.2 together with

their boundary curve data.

We also give the corresponding boundary slopes of the surfaces, were applicable:

Note that in the cases marked with “or ∅”, the possibility exists that the surface corre-

sponding to the coefficients (a, b) does not intersect one of the two cusps. This is the

case when pi = qi = 0 for a valid combination (a, b) 6= (0, 0). In case I, this happens

when one of a and b is zero, and in cases VI or VIII, when a = b. Since (a, b) 6= (0, 0),

the boundary slopes (viewed in Q ∪ {∞}) are well-defined in all other cases.

7The slope pi/qi corresponds to the element µpii λ

qii ∈ im(π1(∂iW ) → π1(W )) (i = 0, 1).

85

Combination p0 q0 p1 q1 p0/q0 p1/q1

I a 3 + b 2 0 −b 0 a 0 or ∅ 0 or ∅

II a 9 + b 8 2a −b −2b a −2ab

−2ba

III a 9 + b 17 2a+ 4b b 2b a+ b 2a+4bb

2ba+b

IV a 8 + b 16 −2b −a− b −2a− 4b −b 2ba+b

2a+4bb

V a 17 + b 14 4a+ 4b a+ b 2a− 2b a− b 4 2 or ∅

VI a 16 + b 13 −2a− 4b −a− b −4a− 2b −a− b 2a+4ba+b

4a+2ba+b

VII a 16 + b 19 −2a+ 2b −a+ b −4a− 4b −a− b 2 or ∅ 4

Table 4.2 Linear combinations of vertex solutions with integers a, b ≥ 0 and their

boundary curves

Now note that in the cases II, III, IV and VI, the boundary slope p1/q1 can be

expressed by the boundary slope r := p0/q0. Furthermore, since a, b ≥ 0, each of the

listed boundary slopes p0/q0 is restricted to a certain interval in Q∪{∞}. We summarize

this data in table 4.3, sorted by range. Note that the symmetry between the boundary

components can clearly be seen from the boundary slopes.

We therefore conclude that for each slope r ∈ Q∪{∞}, there is an essential surface

in Wrh having r as boundary slope on the first torus boundary, and the corresponding

slope given in table 4.3 as boundary slope on the second torus boundary. Additionally,

there are essential surfaces corresponding to the cases I, V and VII which intersect either

one or both boundary components with slope in {0, 2, 4}. In particular, we confirm that

W is a small manifold, i.e. it contains no closed essential surfaces8.

4.1.2 Boundary slopes of the once-filled Whitehead link exterior

Let W = Wrh, and denote its two boundary components by T0 and T1. We now

proceed to determine the boundary slopes of the once-filled Whitehead link exterior

Wp/q = W (p/q, · ) = W ∪µp0λ

q0V , where V is the attached solid torus. To simplify

8Note that a closed incompressible surface can always be normalized. There is in fact one closed

normal surface in W , but it is a genus 2 surface which compresses to two boundary parallel tori.

86

case r = p0/q0 p1/q1

II −2ab ∈ [−∞, 0] 4r−1 = 4q0

p0∈ [−∞, 0]

IV 2ba+b ∈ [0, 2] 4r−1 + 2 = 2p0+4q0

p0∈ [4,∞]

VI 2a+4ba+b ∈ [2, 4] 6− r = −p0+6q0

q0∈ [2, 4]

III 2a+4bb ∈ [4,∞] 4(r − 2)−1 = 4q0

p0−2q0∈ [0, 2]

Table 4.3 Variable boundary slopes of Wrh and their ranges

notation, we will always assume that q > 0, whereas p ∈ Z can be negative. We also

always assume p and q to be coprime.

The main observation which will enable us to find all boundary slopes of Wp/q is

that an essential surface in Wp/q necessarily comes from an essential surface in W : Let

S be an essential surface in Wp/q. We can isotope it so it intersects T0 and T1 ⊂ Wp/q,

as well as the core of V , transversely. If S ∩ T0 = S ∩ T1 = ∅, S can be viewed as a

closed essential surface in W , which is a contradiction, since W is small. We therefore

see that S has to intersect at least one of T0 or T1.

If the intersection S∩ intV is empty, S is in fact an essential surface in W ⊂Wp/q

with non-empty boundary curve on T1. It therefore must arise as a surface in one of the

cases I or VII from the previous section, i.e. its boundary slope on the second boundary

torus is either 0 or 4.

If on the other hand the intersection S∩ intV is non-empty, a standard argument

shows that S ∩ T0 consists of simple closed curves of slope p/q. This implies that

S′ = S ∩ W is an essential surface in W with boundary slope p/q on T0. Unless

p/q ∈ {0, 4}, this implies that S′∩T1 6= ∅, and we get the corresponding boundary slope

on T1 from table 4.3, dependent on p/q.

We therefore have at most three boundary slopes for each manifold Wp/q, which

we summarize in table 4.1.2: We have two fixed slopes β1 = 4 and β3 = 0 and one

varying slope β2 which depends on p and q. The parametrization of the latter depends

on the range of the filling slope p/q. Note that we consider the slopes as elements in

Q ∪ {∞}, where ∞ = 0/1 and −∞ = 0/(−1) are identified.

87

We remark at this point that one could use standard topological methods to

determine when these slopes are indeed boundary slopes in Wp/q. We will not pursue

this here, as we will in fact obtain this from Corollary 4.3.1: We will see in section 4.3

that apart from certain exceptions9, the coefficients of the Culler-Shalen seminorms for

Wp/q associated to these three slopes are strictly positive, which implies that these slopes

are in fact strongly detected (and hence strict) boundary slopes (cf. Lemma 1.2.6). In

summary, we will see:

1. β1 is a strongly detected slope unless p = 2q±1, in which case it is not a boundary

slope.

2. β2 is always a strongly detected slope.

3. β3 is a strongly detected slope when q > 1, and a fibre slope when q = 1.

See section 4.3.5 for details. In the following we will simply use that the set of boundary

slopes of Wp/q is contained in the set {β1, β2, β3}, as this is all we need to proceed

with our calculations.

p/q ∈ β1 β2 β3

[−∞, 0) 4 4qp ∈ (−∞, 0] 0

[0, 2) 4 2p+4qp ∈ (4,∞] 0

[2, 4) 4 −p+6qq ∈ (2, 4] 0

[4,∞) 4 4qp−2q ∈ (0, 2] 0

Table 4.4 Possible boundary slopes for Wrh(p/q) (q > 0)

Remark 4.1.2 When considering distances of β2 to other slopes, it is necessary to

express β2 as a fraction with relatively prime numerator and denominator. This means

that care has to be taken in the following cases: When p/q = 0, we have β2 = ∞ = 1/0.

When p/q = ∞ = 1/0, we get β2 = 0/1. When p is even, a factor of 2 or 4 needs to

9When q = 1, and when p = 2q ± 1, cf. remarks 4.3.2.

88

be cancelled first. Finally, when p is odd, the given parametrization for β3 is already in

relatively prime form.

4.2 The Culler-Shalen seminorm of the Seifert fillings

It is well-known (see e.g. [MP02] or [DT03, §9]) that for σ ∈ {1, 2, 3}, Wp/q(σ) =

Wrh(p/q, σ) is a Seifert fibered manifold with base orbifold a sphere with three cone

points:

• Wrh(p/q, 1) is Seifert fibered over S2(2, 3, |p− 6q|) if p 6= 6q

• Wrh(p/q, 2) is Seifert fibered over S2(2, 4, |p− 4q|) if p 6= 4q

• Wrh(p/q, 3) is Seifert fibered over S2(3, 3, |p− 3q|) if p 6= 3q

This can be seen from the fact that Wrh(1), Wrh(2) and Wrh(3) are all Seifert fibred with

base orbifold a disc with two cone points: D2(2, 3), D2(2, 4) and D2(3, 3), respectively.

Furthermore, the corresponding fibre slopes are 6, 4 and 3, respectively. Therefore, all

but one10 Dehn filling Wrh(1, p/q) ∼= Wrh(p/q, 1) is Seifert fibred with base orbifold a

sphere with three cone points, and similarly for Wrh(2, p/q) and Wrh(3, p/q).

We will use this fact to prove:

Theorem 4.2.1 For p odd, the total Culler-Shalen seminorms of the three Seifert filling

slopes 1, 2 and 3 of Wrh(p/q) are given by

‖1‖ = s+ 2|p− 6q| − 2,

‖2‖ = s+ 3|p− 4q| − 3,

‖3‖ = s+ 4|p− 3q| − 4,

where s is the minimal total Culler-Shalen seminorm.

10The reader can easily verify that the exceptional fillings are Wrh(1, 6) ∼= L2#L3, Wrh(2, 4) ∼=

L2#L4 and Wrh(3, 3) ∼= L3#L3.

89

When p is even, there is a well-known obstruction to lifting PSL2(C) characters

to SL2(C) which needs to be determined for each of the characters in question. We will

address this question in section 4.2.3, but will for the purposes of this work concentrate

here on the case when p is odd.

The proof of Theorem 4.2.1 will be given over the next subsections, and will make

use of the following theorem:

Theorem 4.2.2 ([BAB01, Theorem C])

Let M be the exterior of a knot in a closed, connected, orientable 3-manifold V for which

Hom (π1(V ),PSL2(C)) contains only diagonalizable representations. Suppose that there

is a non-boundary slope r for which M(r) is an irreducible, non-Haken small Seifert

manifold. Fix a non-trivial, irreducible curve X0 ⊂ X(M) for which fα(r)|X0 is non-

constant and let R0 be the component in R which projects to it. Then

‖α(r)‖X0= sX0 + 2A,

where sX0 is the minimal total seminorm and A is the number of characters of non-

abelian SL2(C) representations ρ ∈ R0 such that ρ(α(r)) = ±I. �

Note that in our case M = Wp/q can be considered to lie in V = Wp/q(∞), where

π1(V ) is the finite cyclic group Z/p. It follows that every representation ρ ∈ R(V ) has

a diagonizable image.

To determine the Culler-Shalen seminorm of a Seifert filling slope σ ∈ {1, 2, 3} of

Wp/q, we therefore need to determine the number of characters of non-abelian SL2(C)

representations ρ ∈ R(Wp/q) which satisfy ρ(σ) = ±I, and such that χρ lies on a non-

trivial curve of X(Wp/q). Note that such a representation necessarily induces a (non-

abelian11) PSL2(C) representation ρ ∈ R(Wp/q) with ρ(σ) = ±I, which in turn factors

through to give a representation ρ′ ∈ R(Wp/q(σ)). Since Wp/q(σ) has base orbifold

11An abelian irreducible representation ρ ∈ R necessarily has image isomorphic to Z/2 ⊕ Z/2,

generated by ±`

i 00 −i

´and ±

“0 −b

b−1 0

”for some b ∈ C∗. This possibility is excluded in our situation by

the fact that p is odd.

90

S2(a, b, c), a sphere with three cone points, there is a presentation of π1(Wp/q(σ)) of the

form

π1(Wp/q(σ)) =⟨x, y, h | h central, xa = ha

′, yb = hb

′, (xy)c = hc

′⟩

(4.4)

where gcd(a, a′) = gcd(b, b′) = gcd(c, c′) = 1 (cf. [Jac80, § VI]).

The following lemma from [BAB01] now shows that the non-abelian representa-

tion ρ′ ∈ R(Wp/q(σ)) again factors through to give a representation ρ′′ ∈ R(∆(a, b, c)),

where ∆(a, b, c) ∼= πorb1 (S2(a, b, c) is the (a, b, c) triangle group with presentation

∆(a, b, c) =⟨x, y | xa = yb = (xy)c = 1

⟩: (4.5)

Lemma 4.2.3 ([BAB01, Lemma 3.1])

Let ρ′ ∈ R(Wp/q(σ)). Denote the central element of π1(Wp/q(σ)) corresponding to a

Seifert fibre by h.

1. If ρ′(h) is parabolic, then χρ′ is the character of the trivial representation.

2. If ρ′(h) 6= ±I and is diagonizable, then ρ′(π1(Wp/q(σ))) is diagonizable. �

Since in our situation ρ′ is non-abelian, none of the two options of the Lemma can be

satisfied, and we can hence conclude that ρ′(h) = ±I. This enables us to focus our

attention on the representations of the triangle groups.

We summarize the situation in the following diagram:

π1(Wp/q)ρ

//

��

ρ

&&NNNNNNNNNNNSL2(C)

��

π1(Wp/q(σ))

��

ρ′// PSL2(C)

∆(a, b, c)

ρ′′77ppppppppppp

(4.6)

91

4.2.1 Irreducible PSL2(C) characters

We will first determine the number of irreducible PSL2(C) characters. Let bxc

denote the greatest integer less than or equal to x ∈ R. We will use:

Theorem 4.2.4 ([BAB01, Theorem D])

Let M be a non-Haken small Seifert manifold, with base orbifold of the form S2(a, b, c)

where a, b, c ≥ 2.

1. The number of reducible PSL2(C)-characters of π1(M) is

⌊|H1(M)|

2

⌋+ 1 +

0 if gcd(a, b, c) is even,

1 if gcd(a, b, c) is odd..

2. The number of PSL2(C)-characters of π1(M) representations with image a dihedral

group of order at least 4 is

σ(b, c)⌊a

2

⌋+ σ(a, c)

⌊b

2

⌋+ σ(a, b)

⌊c

2

⌋− 2σ(a, b)σ(a, c)σ(b, c)

where σ(m,n) =⌊

gcd(2,m,n)2

⌋is 1 if m and n are both even, and 0 otherwise.

3. The total number of PSL2(C)-characters of π1(M) is⌊|H1(M)|

2

⌋+ 1 +

⌊a

2

⌋⌊b

2

⌋⌊c

2

⌋+

⌊a− 1

2

⌋⌊b− 1

2

⌋⌊c− 1

2

⌋−

⌊gcd(ab, ac, bc)

2

⌋+

⌊gcd(a, b)

2

⌋+

⌊gcd(a, c)

2

⌋+

⌊gcd(b, c)

2

⌋�

Applying this to the three Seifert fillings of Wrh(p/q), we find the results summarized in

tables 4.5, 4.6 and 4.7, where we made use of the fact that H1(Wp/q(r/s)) = Z/|p|⊕Z/|r|.

We also already included the number of non-abelian reducible characters, which will be

determined in section 4.2.2.

4.2.2 Non-abelian reducible PSL2(C) characters

In addition to the irreducible characters, we need to consider characters of non-

abelian reducible representations of these manifolds, and in particular of the fundamen-

tal groups of their base orbifolds.

92

gcd(

6,p)

tota

lir

redu

cibl

edi

hedr

alre

duci

ble

non-

ab.re

d.

11 2(|p|+

|p−

6q|)

1 2(|p

−6q|−

1)0

1 2(|p|+

1)0

31 2(|p|+

|p−

6q|)

1 2(|p

−6q|−

1)0

1 2(|p|+

1)0

21 2(|p|+

|p−

6q|)

+1

1 2|p−

6q|

11 2|p|+

10

61 2(|p|+

|p−

6q|)

1 2|p−

6q|−

11

1 2|p|+

11

Tab

le4.

5N

umbe

rof

PSL

2(C

)-ch

arac

ters

ofπ

1W

rh(p/q,1

)

93

gcd(

4,p)

tota

lir

redu

cibl

edi

hedr

alre

duci

ble

non-

ab.re

d.

1|p|+

|p−

4q|

|p−

4q|−

11 2(|p

−4q|−

1)|p|+

10

2|p|+

|p−

4q|+

2|p−

4q|

1 2|p−

4q|+

1|p|+

20

4|p|+

|p−

4q|+

1|p−

4q|−

11 2|p−

4q|+

1|p|+

21

Tab

le4.

6N

umbe

rof

PSL

2(C

)-ch

arac

ters

ofπ

1W

rh(p/q,2

)

94

gcd(

6,p)

tota

lir

redu

cibl

edi

hedr

alre

duci

ble

non-

ab.re

d.

13 2(|p|−

1)+|p−

3q|

|p−

3q|−

10

3 2(|p|−

1)+

10

33 2(|p|−

1)+|p−

3q|−

1|p−

3q|−

20

3 2(|p|−

1)+

11

23 2|p|+

|p−

3q|

|p−

3q|−

10

3 2|p|+

10

63 2|p|+

|p−

3q|−

1|p−

3q|−

20

3 2|p|+

11

Tab

le4.

7N

umbe

rof

PSL

2(C

)-ch

arac

ters

ofπ

1W

rh(p/q,3

)

95

Recall that we showed in section 3.1.1 that each character of a non-abelian re-

ducible representation lies on a non-trivial curve and hence contributes to the Culler-

Shalen seminorm.

We will now proceed to determine the number of non-abelian reducible PSL2(C)

characters of π1(Wp/q(α)) for α = 1, 2, 3, by determining the number of such characters

of the triangle groups ∆(a, b, c) (cf. diagram (4.6)).

We will often make implicit use of the following technical lemma:

Lemma 4.2.5 For n ∈ Z and x, y ∈ C∗, we have

x y

0 x−1

n

=

xn y(x2n−1)xn−1(x2−1)

0 x−n

if x 6= ±1,

xn

1 nxy

0 1

if x = ±1,

Proof. Induction on n: The claim is clearly true for n = 0. We first show the induction

step n→ n+ 1. For x = ±1, we havex y

0 x−1

n+1

=

x y

0 x−1

xn

1 nxy

0 1

= xn

x nx2y + x−1y

0 x−1

= xn+1

1 (n+ 1)y

0 1

.

Similarly, for x 6= ±1, we havex y

0 x−1

n+1

=

x y

0 x−1

xn y(x2n−1)xn−1(x2−1)

0 x−n

=

xn+1 xy(x2n−1)xn−1(x2−1)

+ yx−n

0 x−(n+1)

=

xn+1 yx2(x2n−1)+y(x2−1)xn(x2−1)

0 x−(n+1)

=

xn+1 y(x2n+2−1)xn(x2−1)

0 x−(n+1)

.

Multiplying these equations on the left by( x y

0 x−1

)−1 then also gives us the induction

step n+ 1 → n, which proves the claim for all n ∈ Z. �

96

We will now determine the number of PSL2(C) characters of the relevant triangle

groups:

Lemma 4.2.6 The triangle group ∆(2, 3, n) (n 6= 0) admits precisely one non-abelian

reducible PSL2(C) character when 6 | n, otherwise none.

Proof. Let ρ : ∆(2, 3, n) =⟨x, y | x2 = y3 = (xy)n = 1

⟩→ PSL2(C) be a non-abelian

reducible representation. Up to conjugation, we can assume

ρ(x) = ±

i 0

0 −i

and ρ(y) = ±

b 1

0 b−1

,

where b3 = 1, but b 6= 1 (as otherwise ρ were abelian). So we have

ρ(xy)n = (ρ(x)ρ(y))n = ±

(ib)ni((ib)2n−1)

(ib)n−1((ib)2−1)

0 (ib)−n

,

which is equal to ±I iff 6 | n. It now can be seen that the two choices for b give two

representations which are conjugate, and hence have the same character. �

Lemma 4.2.7 The triangle group ∆(2, 4, n) (n 6= 0) admits precisely one non-abelian

reducible PSL2(C) character when 4 | n, otherwise none.

Proof. Let ρ : ∆(2, 4, n) =⟨x, y | x2 = y4 = (xy)n = 1

⟩→ PSL2(C) be a non-abelian

reducible representation. Up to conjugation, we can assume

ρ(x) = ±

a 0

0 a−1

and ρ(y) = ±

b 1

0 b−1

,

where a4 = b8 = 1, but a, b 6= ±1. So in fact a = ±i, and we have b 6= ±i, as

otherwise (ρ(x)ρ(y))n = (ab)n(

1 −na0 1

)6= ±I for n 6= 0, a contradiction. This means

that b = ±e±iπ/4, and we get

ρ(xy)n = (ρ(x)ρ(y))n = ±

anbn a((−1)nb2n−1)an−1bn−1(−b2−1)

0 a−nb−n

,

which is equal to ±I iff 4 | n. As in the previous lemma, it now can be seen that the

choices for b yield conjugate representations and hence precisely one character. �

97

Lemma 4.2.8 The triangle group ∆(3, 3, n) (n 6= 0) admits precisely one non-abelian

reducible PSL2(C) character when 3 | n, otherwise none.

Proof. Let ρ : ∆(3, 3, n) =⟨x, y | x3 = y3 = (xy)n = 1

⟩→ PSL2(C) be a non-abelian

reducible representation. Up to conjugation, we can assume

ρ(x) = ±

a 0

0 a−1

and ρ(y) = ±

b 1

0 b−1

,

where a3 = b3 = 1, but a, b 6= 1 (as otherwise ρ were abelian). Furthermore, b 6= a−1,

as otherwise (ρ(x)ρ(y))n = ± ( 1 na0 1 ) 6= ±I for n 6= 0, a contradiction. So in fact

a = b = e±2πi/3, and we have

ρ(xy)n = (ρ(x)ρ(y))n = ±

(ab)na((ab)2n−1)

(ab)n−1((ab)2−1)

0 (ab)−n

= ±

a2n a4n−1a2n(a−1)

0 a−2n

,

which is equal to ±I iff 3 | n. It now can be seen that all representations of this form

yield conjugate representations and hence precisely one character. �

For future reference, we also study the numbers of SL2(C) characters for these

triangle groups:

Lemma 4.2.9 The triangle groups ∆(2, 3, n) and ∆(2, 4, n) do not admit any non-

abelian SL2(C) representations.

Proof. Let ∆ be either of the two triangle groups, and denote by x a generator of

order two in ∆. Now note that for any representation ρ : ∆ → SL2(C), we can assume

ρ(x) = −I, as this is the only element of order two in SL2(C) up to conjugation. But

this implies that ρ is abelian. �

Lemma 4.2.10 The triangle group ∆(3, 3, n) admits precisely one non-abelian reducible

SL2(C) character when 3 | n, otherwise none.

Proof. Almost identical to the proof of Lemma 4.2.8.

Let ρ : ∆(3, 3, n) =⟨x, y | x3 = y3 = (xy)n = 1

⟩→ SL2(C) be a non-abelian reducible

98

representation. In particular, ρ(x) is diagonizable since ρ(x) 6= ±I. Up to conjugation,

we can therefore assume

ρ(x) =

a 0

0 a−1

and ρ(y) =

b 1

0 b−1

,

where a3 = b3 = 1, but a, b 6= 1 (as otherwise ρ were abelian). Furthermore, b 6= a−1,

as otherwise (ρ(x)ρ(y))n = ( 1 na0 1 ) 6= I for n 6= 0, a contradiction. So in fact a = b =

e±2πi/3, and we have

ρ(xy)n = (ρ(x)ρ(y))n =

(ab)na((ab)2n−1)

(ab)n−1((ab)2−1)

0 (ab)−n

=

a2n a4n−1a2n(a−1)

0 a−2n

,

which is equal to I iff 3 | n. Now note that all these representations have the same

character given by trace ρ(x) = trace ρ(y) = a + a−1 = −1, trace ρ(xy) = a2 + a−2 =

−1. �

We summarize the number of non-abelian reducible SL2(C) and PSL2(C) charac-

ters in table 4.8.

∆(2, 3, n) ∆(2, 4, n) ∆(3, 3, n)

PSL2(C)

1 if 6 | n,

0 otherwise

1 if 4 | n,

0 otherwise

1 if 3 | n,

0 otherwise

SL2(C) 0 0

1 if 3 | n,

0 otherwise.

Table 4.8 Number of SL2(C) and PSL2(C) characters of the base orbifold groups

4.2.3 Lifting the characters

We now use the results of the previous subsections to determine the number of

non-abelian SL2(C) characters for the Seifert fillings.

Note that to do this we need to lift PSL2(C) representations. There is in general

99

an obstruction to this lifting, which is contained in the cohomology group

H2(Wp/q; Z/2) ∼=

0 if p is odd,

Z/2 if p is even.(4.7)

The number of lifts of a given PSL2(C) representation (if they exist) is then generically

given by the cardinality of

H1(Wp/q; Z/2) ∼=

Z/2 if p is odd,

Z/2 ⊕ Z/2 if p is even.(4.8)

Note that dihedral representations only lift half as many times as given by this number.

See e.g. Lemma 5.5 of [BZ96] and the remarks following it for more details.

We therefore have no obstruction to lifting when p is odd. In this case dihedral

PSL2(C) characters are covered by only one SL2(C) character, whereas the remain-

ing non-abelian characters are covered by two. We therefore get the results listed in

table 4.9.

irred. non-dihedral dihedral non-ab. red. total

Wrh(p/q, 1) |p− 6q| − 1 0 0 |p− 6q| − 1

Wrh(p/q, 2) |p− 4q| − 1 12(|p− 4q| − 1) 0 3

2 (|p− 4q| − 1)

Wrh(p/q, 3)

2|p− 3q| − 2 if 3 - p

2|p− 3q| − 4 if 3 | p0

0 if 3 - p

2 if 3 | p2 (|p− 3q| − 1)

Table 4.9 Number of non-abelian SL2(C) characters for Seifert fillings of Wrh(p/q)

when p is odd

Applying Theorem 4.2.1, we now obtain that the difference of the total Culler-

Shalen seminorm of each Seifert filling slope to the minimal total seminorm s is exactly

twice the value given in table 4.9, i.e.

‖1‖ = s+ 2|p− 6q| − 2,

‖2‖ = s+ 3|p− 4q| − 3,

‖3‖ = s+ 4|p− 3q| − 4.

100

This concludes the proof of Theorem 4.2.1.

When p is even, the situation is more complicated since we possibly do have an

obstruction to lifting. In particular, it is to be expected that some representations do

lift, but not others. Of those which do lift, the dihedral representations will lift to

two, and the remaining ones will lift to four SL2(C) representations. To determine the

correct number of SL2(C) representations for p even, one therefore needs to explicitly

determine the corresponding obstruction classes for the relevant representations. This

will be subject of our continuing research in the near future.

4.3 The total Culler-Shalen seminorm of the filled manifolds

Using Lemma 1.2.7, we can now determine the coefficients of the total Culler-

Shalen seminorm from the information obtained from the 1, 2, 3 fillings.

The general strategy is as follows: We know from the previous sections that

‖σ‖ = s+dp,q,σ, where s is the minimal total seminorm, and the dp,q,σ are the differences

obtained for σ ∈ {1, 2, 3} in Theorem 4.2.1.

On the other hand, we know that ‖σ‖ =∑3

i=1 ai∆(σ, βi) and that ‖∞‖ = s.

From these equations, we obtain a linear system with the ai as unknowns which we can

try to solve. In some cases, this will in fact already determine the value for the minimal

total norm — but in most cases, we will also need the expression of s in terms of p and

q as obtained in chapter 3.

Note that depending on the range of the filling slope p/q, the parametrization of

the boundary slope β2 changes (cf. table 4.1.2 on page 87), and hence we get different

distances and accordingly different linear systems for each range. We will therefore

discuss each range in sequence. Furthermore, the seminorms obtained for the Seifert

fillings depend on the parity of p, and so we also need to distinguish between p odd and

p even. Note that we will always assume q > 0 and p ∈ Z with (p, q) = 1. The final

results are summarized in section 4.3.5.

101

4.3.1 Solving for p/q ∈ (−∞, 0), p odd

For p/q ∈ (−∞, 0), p odd, q > 0, we obtain the following distances between the

boundary, Seifert and meridional slopes:

∆ β1 = 4 β2 = 4qp β3 = 0 1 2 3 µ = ∞

β1 0 4(−p+ q) 4 3 2 1 1

β2 0 4q −p+ 4q −2p+ 4q −3p+ 4q −p

β3 0 1 2 3 1

Table 4.10 Distances between slopes in Wrh(p/q) for p/q ∈ (−∞, 0), p odd, q > 0

Using the expressions of the seminorms for the Seifert slopes from Theorem 4.2.1 in

terms of the minimal seminorm s on one hand and the corresponding linear combinations

with the slope distances on the other, we get the following linear system to solve for s,

a1, a2 and a3:

s+ 2(−p+ 6q − 1) = ‖1‖ = 3a1 + (−p+ 4q)a2 + a3 (4.9a)

s+ 3(−p+ 4q − 1) = ‖2‖ = 2a1 + (−2p+ 4q)a2 + 2a3 (4.9b)

s+ 4(−p+ 3q − 1) = ‖3‖ = a1 + (−3p+ 4q)a2 + 3a3 (4.9c)

s = ‖∞‖ = a1 − pa2 + a3 (4.9d)

which is equivalent in matrix form to3 −p+ 4q 1 −1

2 −2p+ 4q 2 −1

1 −3p+ 4q 3 −1

1 −p 1 −1

︸ ︷︷ ︸

A

a1

a2

a3

s

=

2(−p− 1) + 12q

3(−p− 1) + 12q

4(−p− 1) + 12q

0

(4.10)

Since the second row of A is equal to one half of the sum of the first and third rows,

the rank of A is seen to be three, and the system is therefore underdetermined. We

therefore need more information to determine the solution.

If Conjecture 2.6.8 is true (recall that we know this is the case e.g. when q = 1),

we can use the value for s given in table 3.1. In this case, there is a unique solution to

102

the system (4.10), namely: a1

a2

a3

s

=

−p+ 2q − 1

2

2q − 2

−3p+ 4q − 3

. (4.11)

We will now see that we can solve the system (4.10) also without assuming the

validity of Conjecture 2.6.8, which in turn provides a proof of the conjecture for p/q < 0.

4.3.1.1 Using the upper bound for the minimal seminorm

Recall from chapter 3 that the minimal seminorm corresponds precisely to the

number of distinct non-zero roots of the polynomial resp,q. Without using Conjec-

ture 2.6.8, we can therefore only conclude that s ≤ −3p + 4q − 3. In other words, we

have

s = −3p+ 4q − 3− z, (4.12)

where z = z(p, q) ≥ 0 is some positive, integer-valued function in p and q.

Using this expression for s, we can solve the system (4.10) to obtain the following

solution in terms of z: a1

a2

a3

s

=

−p+ 2q − 1− z

2

2 + z4q

2q − 2 + z p−2q4q

−3p+ 4q − 3− z

. (4.13)

We can now use the fact that a1, a2, a3 and s all are even, non-negative integers (cf.

Lemma 1.2.7): The expression for a2 then implies that z is divisible by 8q, i.e. z = 8qz′

for some non-negative integer-valued z′ = z′(p, q). Using this in the expression for a3

then shows that 2q − 2 + 2z′(p− 2q) ≥ 0, which implies that

z′(p− 2q) ≥ 1− q. (4.14)

103

Since p < 0 and q > 0 implies p− 2q < 0, we then get

z′ ≤ 1− q

p− 2q=

q − 12(q + |p|/2)

< 1. (4.15)

But since z′ ∈ Z+, we must conclude that in fact z′ = z = 0.

Note that apart from determining the coefficients of the total seminorm, we have

hereby also proven Conjecture 2.6.8 for p/q < 0.

4.3.1.2 Assuming no knowledge of the minimal seminorm

We would like to point out to what extent the linear system (4.10) can be solved

without assuming any knowledge of the minimal seminorm s.

Even though the system is underdetermined, we can still determine a1, a2 and a3

in terms of s: a1

a2

a3

s

=

12(p+ 1 + s)

14q (−3(p+ 1) + 12q − s)

14q (−3p2 + 10pq − 3p− 2q + (−p+ 2q)s)

s

(4.16)

Under the assumption12 that s is a linear function in p and q, i.e. that s = xp+ yq + z

with x, y, z ∈ Z which do not depend on p and q, we can solve the above system modulo

one unknown integer. In fact, using that a1, a2 and a3 are even non-negative integers,

we can show that

s = −3p+ (4k)q − 3 for a k ∈ Z. (4.17)

Proof. Assume s = xp+ yq + z with x, y, z ∈ Z which do not depend on p and q. We

then get that a2 = 14q (−3(p+1)+12q−s) = 1

4q (p(−3−x)+q(12−y)+(−3−z)). From

the assumption that a2 ∈ Z we thus get that x = 4qx′−3, y = 4y′−12 and z = 4qz′−3

12This is in fact satisfied in our situation, and would be implied by Conjecture 5.4.7 in the general

case, see page 135.

104

for some x′, y′, z′ ∈ Z. But since we assumed that x and z do not depend on q, we must

have x = −3 and z = −3, while y ∈ 4Z, as claimed. (The assumption a3 ∈ Z does not

give any new information.) �

Hence we geta1

a2

a3

s

=

12(p+ 1− 3p+ 4kq − 3)

14q (−3(p+ 1) + 12q + 3p− 4kq + 3)

14q (−3p2 + 10pq − 3p− 2q + (−p+ 2q)(−3p+ 4kq − 3))

s

(4.18)

=

−p+ 2kq − 1

3− k

(1− k)p+ 2kq − 2

−3p+ 4kq − 3

(4.19)

But now a2 = 3− k has to be a non-negative, even integer. This implies k = 1 or k = 3

are the only possibilities. But a2 = 0 would imply that β2 is not a strict boundary slope,

so in fact k = 1 and we obtain the solution found in (4.11) and (4.13).

This shows that the exact value of s is not always necessary to completely de-

termine the Culler-Shalen seminorm, as long as a (rather strong) assumption on its

behaviour in terms of the filling parameters p and q can be made. Notice that this as-

sumption is directly related to the same behaviour of the degree of the polynomials resp,q

from chapter 2. As such, it might be possible to determine such a behaviour in more

general situations, i.e. for manifolds obtained from other 2-component link exteriors by

Dehn filling on one of the boundary tori.

4.3.2 Solving for p/q ∈ (0, 2), p odd

For p/q ∈ (0, 2), p odd, q > 0, we obtain the following distances between the

relevant slopes:

As before, we use the seminorms of the Seifert fillings slopes as obtained in ta-

ble 4.9 on one hand and the corresponding linear combinations with the slope distances

105

∆ β1 = 4 β2 = 2p+4qp β3 = 0 1 2 3 µ = ∞

β1 0 −2p+ 4q 4 3 2 1 1

β2 0 2p+ 4q p+ 4q 4q −p+ 4q p

β3 0 1 2 3 1

Table 4.11 Distances between slopes in Wrh(p/q) for p/q ∈ (0, 2), p odd

on the other to obtain the following linear system to solve for s, a1, a2 and a3:

s+ 2(−p+ 6q − 1) = ‖1‖ = 3a1 + (p+ 4q)a2 + a3 (4.20a)

s+ 3(−p+ 4q − 1) = ‖2‖ = 2a1 + 4qa2 + 2a3 (4.20b)

s+ 4(−p+ 3q − 1) = ‖3‖ = a1 + (−p+ 4q)a2 + 3a3 (4.20c)

s = ‖∞‖ = a1 + pa2 + a3 (4.20d)

Assuming Conjecture 2.6.8 to be true, we have s = p+ 4q− 3, and we obtain the

unique solution a1

a2

a3

s

=

−p+ 2q − 1

2

2q − 2

p+ 4q − 3

. (4.21)

4.3.2.1 Using the upper bound for the minimal seminorm

We now proceed as in the previous case to obtain the solution without assuming

the validity of Conjecture 2.6.8. We therefore can only assume that s ≤ p+ 4q− 3, i.e.

s = p+ 4q − 3− z, (4.22)

where z = z(p, q) ≥ 0 is some positive, integer-valued function in p and q.

Using this expression for s, we can solve the system (4.20) to obtain the following

106

solution in terms of z: a1

a2

a3

s

=

−p+ 2q − 1− qz

2q−p

2− z2(2q−p)

2q − 2− z2

p+ 4q − 3− z

. (4.23)

We can now use the fact that a1, a2, a3 and s all are even, non-negative integers: The

expression for a2 then implies that z is divisible by 4(2q − p), i.e. z = 4(2q − p)z′ for

some non-negative integer-valued z′ = z′(p, q). Using this in the expression for a1 then

shows that −p+ 2q − 1− 4qz′ ≥ 0, which implies that

z′ ≤ −p+ 2q − 14q

= 1− p+ 1 + 2q4q

< 1, (4.24)

so in fact z′ = z = 0.

This shows the correctness of the solution (4.21) without the assumption of Con-

jecture 2.6.8, and thereby proves the Conjecture for 0 < p/q < 2.

4.3.3 Solving for p/q ∈ (2, 4)− {3}, p odd

For p/q ∈ (2, 4), p odd, we obtain the following distances between the relevant

slopes:

∆ β1 = 4 β2 = p−6qq β3 = 0 1 2 3 µ = ∞

β1 0 p− 2q 4 3 2 1 1

β2 0 −p+ 6q −p+ 5q −p+ 4q |p− 3q| q

β3 0 1 2 3 1

Table 4.12 Distances between slopes in Wrh(p/q) for p/q ∈ (2, 4), p odd

As in the previous cases, we use the seminorms of the Seifert filling slopes as

obtained in Theorem 4.2.1 on one hand and the corresponding linear combinations with

the slope distances on the other to obtain the following linear system to solve for s, a1,

107

a2 and a3:

s+ 2(−p+ 6q − 1) = ‖1‖ = 3a1 + (−p+ 5q)a2 + a3 (4.25a)

s+ 3(−p+ 4q − 1) = ‖2‖ = 2a1 + (−p+ 4q)a2 + 2a3 (4.25b)

s+ 4(|p− 3q| − 1) = ‖3‖ = a1 + |p− 3q|a2 + 3a3 (4.25c)

s = ‖∞‖ = a1 + qa2 + a3 (4.25d)

If p/q ∈ (3, 4), this system has rank 4, and we can determine the unique solution without

assuming Conjecture 2.6.8: a1

a2

a3

s

=

p− 2q − 1

4

2q − 2

p+ 4q − 3

. (4.26)

This proves Conjecture 2.6.8 for 3 < p/q < 4.

If p/q ∈ (2, 3), the system has rank 3. If we assume Conjecture 2.6.8, we obtain

the same solution as given in equation (4.26).

4.3.3.1 Using the upper bound for the minimal seminorm

Without assuming the validity of Conjecture 2.6.8, we can only state that s ≤

p+ 4q − 3, i.e.

s = p+ 4q − 3− z, (4.27)

where z = z(p, q) ≥ 0 is some positive, integer-valued function in p and q.

Using this expression for s, we can solve the system (4.25) to obtain the following

solution in terms of z: a1

a2

a3

s

=

−p− 2q − 1− z(p−4q)

2(p−2q)

4− zp−2q

2q − 2− z2

p+ 4q − 3− z

. (4.28)

108

We can now use the fact that a1, a2, a3 and s all are even, non-negative integers: Since

p is odd, the expression for a1 implies that z is divisible by 4(p−2q), i.e. z = 4(p−2q)z′

for some non-negative integer-valued z′ = z′(p, q). Then a2 = 4 − 4z′ ≥ 0 implies that

z′ ≤ 1. Now suppose z′ = 1, but then a1 = −p− 2q − 1− 2(p− 4q) = −3p+ 6q − 1 <

−6q + 6q − 1 = −1 < 0 (since p > 2q), a contradiction. So in fact z′ = z = 0, and

we obtain the same solution as in equation (4.26), thereby proving Conjecture 2.6.8 for

2 < p/q < 3.

4.3.4 Solving for p/q ∈ (4,∞)− {6}, p odd

For p/q ∈ (4,∞), p odd, we obtain the distances between the relevant slopes

shown in table 4.13.

∆ β1 = 4 β2 = 4qp−2q β3 = 0 1 2 3 µ = ∞

β1 0 4p− 12q 4 3 2 1 1

β2 0 4q |p− 6q| 2p− 8q 3p− 10q p− 2q

β3 0 1 2 3 1

Table 4.13 Distances between slopes in Wrh(p/q) for p/q ∈ (4,∞), p odd

As in the previous cases, we use the seminorms of the Seifert fillings slopes as

obtained in Theorem 4.2.1 on one hand and the corresponding linear combinations with

the slope distances on the other to obtain the following linear system to solve for s, a1,

a2 and a3:

s+ 2(|p− 6q| − 1) = ‖1‖ = 3a1 + |p− 6q|a2 + a3 (4.29a)

s+ 3(p− 4q − 1) = ‖2‖ = 2a1 + (2p− 8q)a2 + 2a3 (4.29b)

s+ 4(p− 3q − 1) = ‖3‖ = a1 + (3p− 10q)a2 + 3a3 (4.29c)

s = ‖∞‖ = a1 + (p− 2q)a2 + a3 (4.29d)

If p/q ∈ (4, 6), this system has rank 4, and we can determine the unique solution without

109

assuming Conjecture 2.6.8: a1

a2

a3

s

=

p− 2q − 1

2

2q − 2

3p− 4q − 3

. (4.30)

This also proves Conjecture 2.6.8 for 4 < p/q < 6.

If p/q ∈ (6,∞), the system has rank 3 without any further assumptions. Further-

more, using the upper bound for the minimal seminorm does not completely determine

the system in this case. But now recall from equation (2.41) that nresp,q = nres−p+4q,q.

Furthermore, p/q > 6 if and only if −p/q+4 < −2, and so the simplicity of the non-zero

roots apart from ±1 of nresp,q follows from the simplicity of the roots of nres−p+4q,q,

which we have proven in section 4.3.1.1. We can therefore assume Conjecture 2.6.8 for

p/q ∈ (6,∞). Entering the exact value for s in the system, we then obtain the same

solution as given in equation (4.30).

4.3.5 Summary

The summary for the coefficients of the Culler-Shalen seminorm for the various

ranges for p/q is given in table 4.14 for p odd. Note that the boundaries of the intervals

in the table arise for different reasons: For the slopes 3, 4 and 6, one of the three Seifert

fillings has one cone index equal to 0. On the other hand, the slopes 0, 2 and 4 give the

boundaries of the different parametrizations for the varying boundary slope β2.

We remark that we were able to prove these results (including the precise value

for the minimal total Culler-Shalen norm, cf. table 3.1) without assumption of (and

thereby proving) Conjecture 2.6.8.

As a consequence, we can now determine when a1, a2 and a3 are strictly positive:

First note that a3 = 0 iff q = 1. This corresponds to the fact that in this case Wp/q is

a punctured torus bundle, with fibre the surface with boundary slope 0. The slope 0 is

therefore not a strict boundary slope in this case, and the associated coefficient therefore

110

vanishes identically. Furthermore note that we have a2 = 4 6= 0 for 2 < p/q < 4 and

a2 = 2 6= 0 otherwise. Finally, a1 = 0 iff p = 2q − 1 (in particular p/q ∈ (0, 2)) or

p = 2q + 1 (p/q ∈ (2, 3)). Apart from these exceptions, a1 6= 0.

Since ai > 0 implies that the fundamental polygon B of the Culler-Shalen semi-

norm has a vertex corresponding to βi, Lemma 1.2.6 shows that βi is a strongly detected,

strict boundary slope, i.e. the slope of an essential surface associated to some ideal point

of X(Wp/q).

We summarize:

Corollary 4.3.1 1. β1 is a strongly detected slope of Wp/q unless p = 2q ± 1.

2. β2 is a strongly detected slope of Wp/q.

3. β3 is a strongly detected slope of Wp/q iff q > 1, otherwise it is the slope of a fibre

in the punctured-torus bundle Wrh(p).

Remarks 4.3.2

1. It can be shown by standard topological methods that the slope β1 is in fact not

a boundary slope when p = 2q ± 1.

2. Note that for an integer filling (q = 1), we obtain a once-punctured torus bundle,

where the surface with slope 0 is a fibre (cf. [HMW92]). Using the classification

of essential surfaces in [FH82], it can be seen that there are no other surfaces with

this slope, and we can conclude that the slope 0 is therefore a boundary slope,

but not a strict boundary slope in this case. In fact, these are the only fillings

with fibrations over S1 (this can be seen from the (classical) Alexander polynomial

∆(t) = qt2 + (p− 2q)t+ q of Wp/q, see the proof of Lemma 3.1.5 and [HMW92]).

111

p/q∈

(−∞,0

)(0,2

)(2,3

)(3,4

)(4,6

)(6,∞

)

a1

−p

+2q−

1−p

+2q−

1p−

2q−

1p−

2q−

1p−

2q−

1p−

2q−

1

a2

22

44

22

a3

2q−

22q−

22q−

22q−

22q−

22q−

2

s−

3p+

4q−

3p

+4q−

3p

+4q−

3p

+4q−

33p−

4q−

33p−

4q−

3

Tab

le4.

14T

hepa

ram

eter

sfo

rth

eto

talC

ulle

r-Sh

alen

sem

inor

mofW

rh(p/q

)fo

rp

odd,q>

0

Chapter 5

Fillings of integer fillings of the Whitehead link exterior

It was shown in [HMW92] that the manifolds obtained by integer filling on the

(left-handed) Whitehead link exterior yield an infinite family of once-punctured torus

bundles. Once-punctured torus bundles and their Dehn fillings have been much stud-

ied, see e.g. [Bak89,Bak90,BH96,BPZ87,BMR95,CJR82,FH82,Hat82,HMW92,Jør77,

Mas00,McC87,NR92,Prz83,Prz84,Rei95,RW99,Thu88,Wee85].

In the following we will use the notation RSSm := Wlh(m − 2) for the manifold

obtained by m−2 filling on one boundary component of the left-handed Whitehead link

exterior. We are in particular interested in the closed manifolds RSSm(p/q) obtained by

further Dehn filling on these manifolds. These manifolds are, apart from a finite number

of exceptions, hyperbolic for |m| > 2 and have appeared repeatedly in connection with

some important open questions in 3-manifold topology. A particularly important recent

result was shown by Roberts, Shareshian and Stein in [RSS03], where the authors proved

that for m < −2 odd and p ≥ q ≥ 1 with p odd, any action of the fundamental

group of RSSm(p/q) on an R-order tree1 is trivial, i.e. has a global fixed point. As

a consequence these manifolds were shown not to contain a Reebless foliation, or a

transversely oriented essential lamination. Furthermore, Fenley showed in [Fen02] that

the manifolds RSSm(p/q) for m ≤ −4 and |p − 2q| = 1 do not admit any essential

laminations, answering in the negative the long-standing question whether all closed

1a simply-connected, second countable but not necessarily Hausdorff 1-manifold

113

hyperbolic 3-manifolds admit essential laminations2.

We will show in this chapter that many of the manifolds RSSm(p/q) nevertheless

virtually admit closed essential surfaces, i.e. they possess finite covers which have this

property. In fact, we will show the stronger result that for p/q away from certain infinite

strips in the surgery planeH1(∂ RSSm; R), the manifolds RSSm(p/q) have positive virtual

first Betti number, confirming the following variant of a conjecture of Waldhausen:

Conjecture 5.0.3 If M is a compact, connected, P2-irreducible 3-manifold with infinite

fundamental group, then there is a finite cover M →M with b1(M) > 0.

We will achieve this using a result of Joseph Masters which, under certain conditions on

the monodromy and the filling slope, gives a construction of finite covers of the fillings

of a once-punctured torus bundle (cf. [Mas00]).

This chapter is organized as follows: In section 5.1, we will introduce some no-

tation and discuss some general properties of once-punctured torus bundles and their

monodromies as well as the choice of a framing on the boundary torus. In section 5.2,

we will then briefly discuss the once-punctured torus bundle structure on the manifolds

RSSm and find various decompositions of their monodromies. In section 5.3, we will

discuss some results about the closed manifolds obtained by Dehn filling the manifolds

RSSm. Finally, in section 5.4, we will apply the work of Masters (cf. [Mas00]) to the

manifolds RSSm(p/q), showing that “most” of them have positive virtual first Betti

number.

The author would like to thank Ian Agol for pointing out Masters’ work on once-

punctured torus bundles.

5.1 Once-punctured torus bundles and their monodromies

We summarize some general results on once-punctured torus bundles, following

the exposition in [CJR82], but using the convention of [Mas00] for the generators (cf.

figure 5.1).

2We remark that this was conjectured for m odd in [RSS03].

114

A once-punctured torus bundle is the surface bundle over S1

Mh := T ∗ × I/ ((x, 0) ∼ (h(x), 1)) , (5.1)

where T ∗ is a compact orientable surface of genus one with single boundary component

and h ∈ Homeo(T ∗), the mapping class group3 of T ∗. Denote by Homeo+(T ∗) the sub-

group of isotopy classes of orientation-preserving homeomorphisms. A classical result of

Nielsen (cf. [Nie17]) shows that Homeo(T ∗) ∼= GL2(Z), with the subgroup of orientation-

preserving homeomorphisms Homeo+(T ∗) corresponding to SL2(Z) ⊂ GL2(Z): Choose

two simple closed curves x and y with intersection number 1 as a basis for π1(T ∗).

We will follow the convention used in [Mas00] for these generators, cf. figure 5.1. For

h ∈ Homeo(T ∗), let h∗ denote the induced automorphism of H1(T ∗) ∼= Z⊕Z. The map

h 7→ h∗ then induces an isomorphism from Homeo(T ∗) to Aut(Z⊕ Z). This last group

can be identified with GL2(Z) by choosing the homology classes of x and y as a basis

for H1(T ∗). The image of h∗ under this identification (or sometimes h itself) is called

the monodromy of the bundle.

h

x

y

λ

x0 tγ

Figure 5.1 Conventions for the once-punctured torus bundle Mh

3The group of homeomorphisms modulo the subgroup of homeomorphisms that are isotopic to

the identity. This is also sometimes called the homeotopy group.

115

Convention 5.1.1 We follow the convention of both [CJR82] and [Mas00] and

consider all arising groups to act on the left on a given element. In particular, we

consider a monodromy matrix h∗ ∈ SL2(Z) to act by left-multiplication on a column

vector representing a homology class, e.g.m −1

1 0

1

0

=

m1

.

In contrast, we remark that both [RSS03] and [Fen02] write all group actions on the

right.

The group Homeo+(T ∗) can be generated by Dehn twists Dx and Dy about the

curves x and y, respectively. Their isotopy classes then correspond to the matrices in

SL2(Z)

R =

1 1

0 1

and L =

1 0

1 1

, (5.2)

respectively. Furthermore, consider the matrices

P =

0 −1

1 0

, and Q =

1 1

−1 0

. (5.3)

We will often make implicit use of the following equalities4:

L = PQ, R = PQ2 and P = LR−1L, Q = L−1R. (5.4)

Note that P 2 = −I = Q3 lies in the center and hence can always be moved to the front

of a word in SL2(Z), as illustrated in

L−1 = (PQ)−1 = Q−1P−1 = Q5P 3 = −Q2P 3 = Q2P, (5.5)

R−1 = (PQ2)−1 = Q−2P−1 = −Q4P = QP. (5.6)

4Note that even though −I = (RL−1R)2 (this is e.g. used in [Mas00]), it should be noted that

P = LR−1L 6= RL−1R. In fact we also have −I = (LR−1L)2 = (RL−1R)2 = (L−1R)3, etc.

116

It then follows that P and Q give the well-known presentation of SL2(Z) as the free

amalgamated product

SL2(Z) = Z4 ∗Z2 Z6 =⟨P,Q | P 4 = Q6 = 1, P 2 = Q3

⟩. (5.7)

The normal form theorem for free products with amalgamation now shows that each

element h∗ ∈ SL2(Z) can be written uniquely as

h∗ = P rX1X2 · · ·Xs, (5.8)

where r ∈ {0, 2} and the Xi are chosen alternately from the two sets {P} and {Q, Q2}.

We call this the (PQ) normal form for h∗. Since P 2 = −I, we will usually simply write

± instead of P r.

Furthermore, h∗ is conjugate to an element g′ whose normal form

h∗ ∼ h′∗ = ±Y1Y2 · · ·Yt (5.9)

has the additional property that Y1 and Yt are not both contained in the same one of

the two sets {P} and {Q, Q2}. This form is obtained by cyclically reducing the normal

form for h∗, and is hence called the cyclically reduced normal form for h∗. It is unique

up to cyclic permutation of Y1 · · ·Yt. The number l(h∗) := t is called the (PQ) syllable

length of h∗. It is 0 iff h∗ = I or h∗ = P 2 = −I; when l(h∗) = 1, then h∗ is conjugate

to one of P±1, Q±1 or Q±2; when l(h∗) > 1, it is even. The conjugacy class of h∗ is

therefore determined by a word5

±PQε1PQε2 · · ·PQεs (5.10)

up to cyclic permutation of the exponents (ε1, . . . , εs), where εi ∈ {1, 2}. Following

[CJR82], we use the notation ±[ε1, . . . , εs] to denote this word.

From PQ = L and PQ2 = R, it now can be seen that up to conjugation, equa-

tion (5.10) gives rise to a product of the matrices L and R of the form

h∗ = ±Ra1Lb1 · · ·RakLbk , (5.11)

5Note that we choose to start the word with a P term, contrary to [CJR82]. This has the

advantage of directly showing the canonical LR decomposition, cf. equation (5.11).

117

where the ai, bi are positive integers. We will call this the canonical (RL) form of h∗.

Any other decomposition of h∗ as a product of the matrices L and R (for example with

some negative exponents) will be called non-canonical. If we want to emphasize that a

certain decomposition is in fact equal (rather than only conjugate) to h∗ ∈ SL2(Z), we

will call it an exact decomposition.

5.1.1 Classification of punctured-torus bundles

Two fibre bundles with the same fibre and base are said to be weakly equivalent if

there is a fibre-preserving homeomorphism between them. They are equivalent if there is

a fibre-preserving homeomorphism between them that induces the identity on the base.

By the classification of fibre bundles over spheres, it is known that the equivalence classes

of bundles over S1 with fibre T ∗ are in one-to-one correspondence with conjugacy classes

in Homeo(T ∗) ∼= GL2(Z). The conjugacy class of h∗ ∈ SL2(Z) corresponding to the

bundle Mh is called its characteristic class and is denoted by [h∗] (cf. [CJR82]).

Theorem 5.1.2 (Murasugi, cf. [CJR82, Proposition 1.3.1]) Once-punctured torus bun-

dles Mh and Mg are homeomorphic if and only if their fibre bundle structures are weakly

equivalent, i.e. if [h∗] = [g±1∗ ]. �

Note that if h∗ ∼ g−1∗ , then Mh

∼= Mg via a fibre-preserving, but orientation-

reversing homeomorphism.

5.1.2 Framings and the fundamental group

The following material can be found in Section 6.2 of [CJR82].

To be able to discuss Dehn fillings on a once-punctured torus bundle M , we need

to fix a framing6 on its boundary ∂M .

First note that there is a unique (up to isotopy) simple closed curve in ∂M given

by the boundary of the fibre T ∗. This curve can be seen as the analogue of the canonical

6The choice of a pair of transverse simple closed curves.

118

longitude7 in a knot exterior in S3 and will be denoted by λ.

Contrary to the situation of a knot exterior in S3, there is no natural choice for

a “meridian”, and therefore some care is required. Fix a base point x0 on ∂T ∗ and let

a and b be generators of π1(T ∗, x0) = F2 (the free group on two generators) such that8

[a, b] = aba−1b−1 = λ. Let Stab([a, b]) be the subgroup of the group of automorphisms

of π1(T ∗, x0) that stabilizes [a, b]. We will see that an element γ ∈ Stab([a, b]) uniquely

determines a framing for ∂M .

First note that for γ ∈ Stab([a, b]), there is a unique (up to isotopy fixing x0)

homeomorphism h ∈ Homeo(T ∗) which induces h] = γ on the fundamental group.

Furthermore, if M = Mh, then up to isotopy fixing x0 there is a unique oriented,

simple closed curve tγ in ∂M which is transverse to the fibre in M and which satisfies

t−1γ atγ = γ(a) and t−1

γ btγ = γ(b). This curve tγ together with λ = ∂T ∗ then determines

a framing for ∂M . We therefore also call the element γ ∈ Stab([a, b]) a framing for h.

We will restrict our attention to the setup of [Mas00]. Note that with the choice

of the generators x and y given in figure 5.1, the boundary curve λ is given by λ =

x−1yxy−1. Assume that M = Mh with a given decomposition of the monodromy h∗ as

a product of the matrices L and R:

h∗ = P rRa1Lb1 · · ·RakLbk . (5.12)

One can now assign a standard choice of an element h] ∈ Stab(λ) and hence of a framing

for ∂M . This can be done by choosing standard framings for the Dehn twists along the

curves x and y, e.g. (following [Mas00]):

Dx] : x 7→ x, y 7→ yx (5.13)

Dy] : x 7→ yx, y 7→ y. (5.14)

7The unique null-homologous curve (up to isotopy) intersecting the meridian in a single point.

8Note that in the convention of [Mas00], the boundary of T ∗ is in fact given by the word

x−1yxy−1. One can choose e.g. a := x−1 and b := y to obtain a more standard pair of generators.

119

Note that Dx] and Dy] are in Stab(λ), and that this choice is not canonical. Using

P = LR−1L and hence P] := Dy]D−1x] Dy], one can then assign to each decomposition

of the form (5.12) the element h] ∈ Stab(λ) given by

h] := P r] Da1x]D

b1y] · · ·D

akx]D

bky] . (5.15)

With this choice of framing, we then obtain the following presentation of π1(Mh) as an

HNN-extension over the free group π1(T ∗) = 〈x, y 〉:

π1(Mh) =⟨x, y, t | t−1xt = h](x), t−1yt = h](y)

⟩, (5.16)

where t = th]. Equivalently, we can write this presentation in terms of the more standard

generators a := x−1 and b := y so that λ = [a, b]:

π1(Mh) =⟨a, b, t | t−1at = h](a), t−1bt = h](b)

⟩. (5.17)

We remark at this point that following Convention 5.1.1, we consider the auto-

morphism h] to act on the left of a given word γ ∈ π1(T ∗, x0).

5.2 Integer fillings of the Whitehead link exterior and their mon-

odromies

It was shown in [HMW92] that for m ∈ Z, the integer filling Wlh(m − 2, · ) on

one component of the left-handed Whitehead link exterior yields a once-punctured torus

bundle with the monodromy9

h∗ =

m −1

1 0

. (5.18)

In particular, at the very end of the article, the authors show a beautiful figure illus-

trating the fibration. We will not try to reproduce this figure here, but rather refer

the reader to the original (cf. [HMW92, Figure 10]). We will denote this manifold by

RSSm := Wlh(m− 2, · ).

9Note that they give the transposed matrix of the one shown here, as they use a right action for

the action of monodromies on H1(T∗).

120

It will turn out that for our applications, we need a non-canonical decomposition

of this monodromy. For completeness and future reference, we give the various canonical

and non-canonical decompositions of this monodromy and its transposed matrix in

table 5.1. We remark that there is a small typographic error in [HMW92] in the note

after Proposition 3, giving an incorrect canonical form in the negative trace case.

monodromy

m −1

1 0

m 1

−1 0

various exact

decomposi-

tions

Rm−1LR−1,

−Rm+1L−1R,

LR−1L1−m,

− L−1RL−m−1

L−1RLm−1,

− LR−1L1+m,

R1−mL−1R,

−R−m−1LR−1

canonical form

(m ≥ 2)Rm−2L RLm−2

canonical form

(m ≤ −2)−RL−m−2 −R−m−2L

cyclically re-

duced normal

form (m ≥ 2)

PQ(PQ2)m−2

= [1, 2, . . . , 2︸ ︷︷ ︸m−2

]

PQ2(PQ)m−2

= [2, 1, . . . , 1︸ ︷︷ ︸m−2

]

cyclically

reduced nor-

mal form

(m ≤ −2)

−PQ2(PQ)−m−2

= −[2, 1, . . . , 1︸ ︷︷ ︸−m−2

]

−PQ(PQ2)−m−2

= −[1, 2, . . . , 2︸ ︷︷ ︸−m−2

]

non-canonical

forms for

m ≤ −2

R−1L2−m,

Rm−2L

R2−mL−1,

RLm−2

Table 5.1 Various decompositions of the monodromy of RSSm

One sees immediately from the cyclically reduced PQ-normal forms that the two

matrices are conjugate in SL2(Z) if and only if m = ±3. This corresponds to the fact

121

that RSS3 (the exterior of the figure-8 knot) and RSS−3 (the so-called sister manifold

of the figure-8) are amphichiral manifolds, i.e. they admit an orientation-reversing self-

homeomorphism.

5.2.1 Inversion and weak conjugacy classes of monodromies

We make a few remarks regarding the inversion of monodromy matrices and their

respective conjugacy classes.

First note that the matrices R and L−1 (or L and R−1) are conjugate in many

ways, e.g.:

R = PQ2 ∼ Q−1(PQ2)Q = −Q2PQ3 = Q2P = L−1,

R−1 = QP ∼ Q−1(QP )Q = PQ = L,

L = PQ ∼ Q(PQ)Q−1 = QP = R−1,

L−1 = Q2P ∼ Q(Q2P )Q−1 = Q3P (−Q2) = PQ2 = R.

In fact, we have more generally:

Lemma 5.2.1 Ra1Lb1 · · ·RakLbk ∼ L−a1R−b1 · · ·L−akR−bk .

In other words: replacing each R by L−1 and each L by R−1 without changing the

order keeps the conjugacy class fixed.

Proof. For k = 1 we get:

Ra1Lb1 = (PQ2)a1(PQ)b1 = P (Q2P )a1(QP )b1P−1 = P (L−a1R−b1)P−1

We thus get for any k ∈ N:

Ra1Lb1 · · ·RakLbk = P (L−a1R−b1)P−1 · · ·P (L−akR−bk)P−1

= P (L−a1R−b1 · · ·L−akR−bkL−a1)P−1.

�

122

This means in particular that the inverse of a word with only two syllables10 lies

in the same conjugacy class as the word obtained by exchanging the exponents:

(RaLb)−1 ∼ RbLa. (5.19)

Considering that cyclic permutations of the exponents by even distances obviously

keeps the conjugacy class fixed, one might be tempted to assume that odd-distance cyclic

permutations will always switch to the inverse conjugacy class — but this in general not

true, as illustrated in the following example: Let w = RL2R2L. Cyclically permuting

the exponents one step to the left yields the word w = R2L2RL which has cyclically

reduced normal form [2, 2, 1, 1, 2, 1]. On the other hand, w−1 has the normal form

[2, 2, 1, 1, 2, 1], which shows that w−1 � w.

5.2.2 The fundamental group

Using the exact decomposition h∗ = LR−1L1−m ∼ L1−mR−1L (cf. table 5.1) and

the corresponding framing h] := D1−my] D−1

x] Dy] ∈ Stab(λ), one finds for m ≤ −2 that

h](x) = y and h](y) = yx−1ym−1. (5.20)

Similarly, for m ≥ 2, the exact decomposition h∗ = Rm−1LR−1 ∼ R−1LRm−1 with the

framing h] := D−1x] Dy]D

m−1x] yields the same result as in equation (5.20).

Using a := y and b := x−1, the fundamental group of RSSm can therefore be

presented as in equation (5.17) by

π1(RSSm) =⟨a, b, t | t−1at = abam−1, t−1bt = a−1

⟩. (5.21)

Here [a, b] = λ, the boundary of the punctured torus fibre, and t = th]is the unique

curve on ∂ RSSm determined by the framing h].

Now note that any element of Stab(λ) conjugate to h] determines the same fram-

ing as h]. This means in particular that the above presentation and its framing given

by t and λ is also valid for h′∗ = R−1L2−m (m ≥ −2) or h′∗ = Rm−2L (m ≥ −2). This

will be important when we apply Theorem 5.4.4, cf. remark 5.4.5 2.

10The number of maximal terms of the form Rai or Lbi

123

5.3 Closed manifolds obtained by Dehn filling the integer fillings

We now consider the manifolds obtained by Dehn filling RSSm with slope (p, q) ≡pq ∈ Q ∪ {∞} with respect to the framing (t, λ). This has the effect of making the

homotopy class tpλq trivial and hence changes the presentation of the fundamental

group to

π1 (RSSm(p/q)) =⟨a, b, t | at = abam−1, bt = a−1, tp = [a, b]−q

⟩. (5.22)

Our interest in these manifolds and their fundamental groups is mainly due to the

following result:

Theorem 5.3.1 (cf. [RSS03, Theorem 2.1 and Corollary 2.3]) Let T be a simply-connected

(second countable, but not necessarily Hausdorff) 1-manifold. Then the image of any

homomorphism Φ: π1 (RSSm(p/q)) → Homeo(T ) has a global fixed point for m < −2

odd, p ≥ q ≥ 1 and p odd. In other words, any action of π1 (RSSm(p/q)) on T is trivial

in these cases. �

As a consequence, the authors proceed to show that the manifolds RSSm(p/q)

satisfying the conditions of Theorem 5.3.1 do not admit a Reebless foliation or a trans-

versely oriented essential lamination. Though most of these manifolds do contain an

essential lamination, the authors conjectured that the fillings with |p − 2q| = 1 and

m < −2 do not admit any essential laminations. This conjecture has in the meantime

been proven by Fenley for m < −3 (both odd and even):

Theorem 5.3.2 ([Fen02]) The manifold RSSm(p/q) does not admit essential lamina-

tions if m ≤ −4 and |p− 2q| = 1. �

It is a natural question to ask whether these manifolds at least virtually admit

essential laminations, i.e. if there exists a finite cover with this property. We will show

in the next section that apart from a finite number of exceptions this is indeed the case.

Remark 5.3.3 At the end of section 2 in [RSS03], the authors ask the question whether

their condition that m be odd is necessary to:

124

1. rule out nontrivial group actions, or to

2. rule out the existence of Reebless foliations.

We would like to point out that the answer to 2 is trivially yes (i.e. the condition

that m be odd is needed) if q = 1: Viewing the manifolds RSSm as integer fillings

on the left-handed Whitehead link W := Wlh, we have RSSm(p/q) = Wm−2(p/q) =

W (m− 2, p/q) = W (p/q,m− 2) = Wp/q(m− 2), by the symmetry of the two boundary

tori of W . If q = 1, this manifold can be seen as Wp/1(m − 2) = RSSp+2(m − 2).

Since the authors remark that RSSm(p/q) contains a Reebless foliation when p is even,

it follows that the same is true for the manifold RSSp+2(m− 2) when m− 2 (and hence

m) is even.

5.4 Applying a result by Masters

Our goal is to apply Theorem 1.3 of [Mas00] (cf. Theorem 5.4.4 below) to show

that most of the manifolds RSSm(p/q) have positive virtual first Betti number, i.e. there

is a finite cover M of RSSm(p/q) such thatH1(M ; Z) is infinite. We will call the manifold

(or its fundamental group) virtually Z-representable in this case. Our main result is:

Theorem 5.4.1 Consider the manifolds RSSm(p/q) for relatively prime integers p, q,

and where m < 0 satisfies:

1. m ≤ −4, m even or 7 | 2−m. Then if |q| > 1, all but finitely many Dehn fillings

RSSm(p/q) are virtually Z-representable.

2. m ≤ −3 and odd, 7 - 2−m. Then if 1|p+q| +

1|p+2q| +

1|q| < 1, RSSm(p/q) is virtually

Z-representable.

3. 4 | 2−m. Then if 2|p+2q| +

1|q| < 1, RSSm(p/q) is virtually Z-representable.

Similarly, for m > 0:

125

4. m ≥ 8, m even or 7 | m − 2. Then if |q| > 1, all but finitely many Dehn fillings

RSSm(p/q) are virtually Z-representable.

5. m ≥ 7 and odd, 7 - m− 2. Then if 1|p−q| +

1|p−2q| +

1|q| < 1, RSSm(p/q) is virtually

Z-representable.

6. 4 | m− 2. Then if 2|p−2q| +

1|q| < 1, RSSm(p/q) is virtually Z-representable.

Before we begin the proof, we discuss some immediate consequences: Since an

essential surface is a special case of an essential lamination, we immediately get that

the manifolds RSSm(p/q) virtually admit an essential lamination in all the applicable

cases of the Theorem. In particular, we can resolve this question for all manifolds from

Theorem 5.3.2:

Corollary 5.4.2 All but possibly a finite number of the manifolds RSSm(p/q) with m ≤

−4 and |p−2q| = 1 are virtually Z-representable and hence virtually possess an essential

surface.

Proof. Let p = 2q±1 and without loss of generality q > 0. In case 1 of Theorem 5.4.1

there are at most a finite number of exceptions, so there is nothing to prove. In case 2,

we get the condition

13q ± 1

+1

4q ± 1+

1q

=19q2 ± 9q + 1q(12q2 ± 7q + 1)

< 1, (5.23)

and this is satisfied iff q > 1. So only p/q ∈ {1, 3} are possible exceptions in this case.

In case 3, we get the condition 24q±1 + 1

q = 6q±1q(4q±1) < 1 which is satisfied iff q > 1. Again,

only p/q ∈ {1, 3} are possible exceptions. �

We also remark that apart from the integer fillings, all manifolds from Theo-

rem 5.3.1 are covered in cases 1 and 2 of Theorem 5.4.1:

Corollary 5.4.3 All manifolds RSSm(p/q) with m < −2 odd and p ≥ q > 1 with p

odd are virtually Z-representable. In particular, their fundamental groups are virtually

left-orderable and hence act non-trivially on an R-order tree.

126

Proof. Simply note that the only non-integer slopes in the first quadrant which are

not covered by Theorem 5.4.1 are (1, 0) and (1, 2), cf. figure 5.2. But these are excluded

by the condition p ≥ q. For the virtual left-orderability note that the existence of a

non-trivial homomorphism from π1(RSSm(p/q)) to Z (a left-orderable group) implies

that π1(RSSm(p/q)) is left-orderable itself (cf. [BRW05, Theorem 3.2]). �

Let us now discuss how Masters’ result applies in our case to prove Theorem 5.4.1.

Theorem 5.4.4 ([Mas00], theorem 1.3) Let Mh be a once-punctured torus bun-

dle over S1 with monodromy h∗ = Da1x D

b1y · · ·Dat

x Dbty , and let A = gcd(a1, . . . , at),

B = b1 + · · ·+ bt.

Suppose A is divisible by some k such that

1. k ≥ 6 even or k = 7. Then if |q| > 1, all but finitely many Dehn fillings

Mh(p/q) are virtually Z-representable.

2. k >= 5 odd, k 6= 7. Then if 1|Bp−q| + 1

|Bp−2q| + 1|q| < 1, Mh(p/q) is virtually

Z-representable.

3. k = 4. Then if 2|Bp−2q| +

1|q| < 1, Mh(p/q) is virtually Z-representable. �

Remark 5.4.5 1. Analogous results hold if we replace A by gcd(b1, . . . , bt), and B

by a1 + · · ·+ at. This holds since the “rows” in Masters’ cover construction can be

applied to either of the x or y directions (cf. [Mas00] for more details).

2. The filling slopes are relative to the framing determined by h] := Da1x]D

b1y] · · ·D

atx]D

bty],

using the standard choices for Dx] and Dy] from equation (5.13). In our case, we

therefore have the framing corresponding to the presentation of the fundamental

group as in equation (5.22).

127

We will now proceed to prove Theorem 5.4.1. All we need to do is find an appro-

priate decomposition of the monodromy of RSSm, to which Masters’ theorem applies.

First note that for m < −2, the canonical decomposition of the monodromy

(cf. table 5.1) is −RL−m−2 = P 2RL−m−2 = (LR−1L)2RL−m−2, and applying Theo-

rem 5.4.4 to this decomposition is impossible, as the gcd of either the L or R exponents

is 1, violating the hypothesis11.

On the other hand, the non-canonical decomposition

h∗ = LR−1L1−m ∼ R−1L2−m = D−1x D2−m

y (5.24)

does permit us to apply Masters’ theorem: We can set A = 2 −m and B = −1 (using

Remark 5.4.5 (1)), and we get the claim for the cases 1, 2 and 3. Note that instead of

assuming that 2−m is divisible by a k, we simply assume 2−m = k in most cases, as

there is no gain in doing otherwise. An exception to this rule is when 4 | 2 −m (case

3) or when 7 | 2 −m. In this last case, the proof of [Mas00, Theorem 1.3] enables us

to be even more precise in determining the remaining “bad” slopes (for which Masters’

theorem doesn’t apply): They lie in the region defined by

(f1 ≤ 0 ∨ f2 ≤ 0) ∧ (g1 ≤ 0 ∨ g2 ≤ 0), (5.25)

where

f1 = 2/3− (1/|p+ q|+ 1/|q|), f2 = |(p+ 2q)2 − 2q2| − 2,

g1 = 2/3− (1/|p+ 2q|+ 1/|q|), g2 = |(p+ q)2 − 2q2| − 2.

The slopes in this region are the integer slopes (q = 1) and the slopes in the finite set

{(1, 0); (−7, 2); (−5, 2); (−3, 2); (−1, 2); (1, 2); (−5, 3); (−4, 3)}.

We show the areas with bad slopes for m < 0 in figures 5.2, 5.3 and 5.4.

For m ≥ 2, we can simply use the canonical form Rm−2L and set A = m− 2 and

B = 1. We then get similar results, with the regions of “bad” slopes shown in figures

11The minus sign in the canonical form corresponds to the negative trace of the matrix, necessary

because of the strictly positive exponents in the product.

128

5.5, 5.6 and 5.7. When 7 | m− 2, the non-integral slopes remaining as bad slopes are in

the set {(1, 0); (−1, 2); (1, 2); (3, 2); (5, 2); (7, 2); (4, 3); (5, 3)}.

0

10

20

30

40

q

–40 –20 20 40

p

Figure 5.2 The bad slopes for m ≤ −3 odd, 7 - 2−m

5

10

15

20

q

–40 –20 20 40

p

Figure 5.3 The bad slopes for m < 0 and 4 | m

129

0

0.5

1

1.5

2

2.5

3

–20 –10 10 20

Figure 5.4 The bad slopes for m < 0 when 7 | 2−m

0

10

20

30

40

q

–40 –20 20 40

p

Figure 5.5 The bad slopes for m ≥ 7 odd, 7 - m− 2

130

5

10

15

20

q

–40 –20 20 40

p

Figure 5.6 The bad slopes for m > 0 when 4 | m− 2

0

0.5

1

1.5

2

2.5

3

–20 –10 10 20

Figure 5.7 The bad slopes for m > 0 when 7 | m− 2

131

We conclude this chapter by putting our result in relationship with the following

result by Dunfield and Thurston12, which is obtained with quite different methods13:

Theorem 5.4.6 ([DT03, Theorem 9.1]) Let W := Wlh be the exterior of the left-handed

Whitehead link in S3, with its canonical framing (cf. figure 1.1 on page 24). For any

slope p/q which is not in the set E = {∞, 0, −1, −2, −3, −4, −5, −5/2, −6, −7, −7/2,

−8, −8/3, −9/2, −10/3, −11/2, −11/3, −13/3, −13/4, −14/3, −15/4, −16/3, −16/5,

−17/5, −18/5, −19/4, −24/5, −24/7} the manifold W (p/q) has the property that all

but finitely many Dehn fillings W (p/q, r/s) have virtual positive first Betti number. �

Because of the symmetry of the two boundary components of W , there are two possi-

bilities to apply this result to our situation:

1. By setting r/s = m− 2, we obtain that for all but finitely many choices of m ∈ Z,

the manifold W (p/q,m− 2) ∼= RSSm(p/q) has virtual positive first Betti number

for all p/q 6∈ E.

2. By setting p/q = m−2, we obtain that for allm 6∈ {−6,−5,−4,−3,−2,−1, 0, 1, 2},

all but finitely many fillings W (m−2, r/s) ∼= RSSm(r/s) have virtual positive first

Betti number.

Note that in the first case, no precise statement about the exceptional m ∈ Z is made

— whereas in the second case, no precise statement about the exceptional slopes r/s is

possible.

In contrast, Theorem 5.4.1 applies to all m ≤ −2 and m ≥ 6 and gives precise

statements about the slopes yielding manifolds with virtual positive first Betti number.

Even though our methods used cannot deal with some infinite bands in the surgery

12They stated this result for the right-handed Whitehead link Wrh, while we restate it here for

Wlh.

13They study one-relator quotients of 2-orbifold groups and use some deep number theoretic

results.

132

plane, the explicit confirmation of “good” filling slopes (in particular with respect to

Corollaries 5.4.2 and 5.4.3) show the relevance of our results.

We finally remark that the slopes in Theorem 5.4.6 cannot directly be compared

with the slopes in our results, as the former are given with respect to the canonical

framing of W as a link exterior, while the latter are given with respect to the “standard”

framing of RSSm as a punctured-torus bundle.

Conclusion

To conclude this thesis, we would like to make some comments regarding the

results of this thesis, and possible generalizations and extensions of these results.

As we have at the moment only determined the Culler-Shalen seminorms for odd

values of p, the natural plan for the near future will of course be to extend these results

to the case when p is even. As all the necessary pieces for this calculation are already

in place in chapter 4, it is only a question of determining the obstruction class for a

given PSL2(C) representation ρ ∈ R(Wp/q) to decide whether it lifts to a SL2(C). We

have begun work on this already and we will concentrate on completing this in the near

future, ultimately aiming for a complete treatment of all but a finite number of the

manifolds Wp/q for arbitrary p.

Another direction for future work is Conjecture 2.6.8 or rather Conjecture 2.6.10.

It seems that there should be an analytical proof of this conjecture, which so far has

evaded a direct approach. Apart from the intrinsic value of such a result by itself, we

could then immediately conclude the exact values for the minimal Culler-Shalen norm

in chapter 3 without using the results of chapter 4.

Finally, it is our belief that the techniques of chapter 2 should generalize to other

2-component link exteriors (at the very least to other 2-bridge links of this type), possibly

under some restrictions (e.g. a hyperbolic link exterior whose once-fillings are all small

manifolds). We will describe how we believe the general procedure should proceed to

obtain the set of p-reps in this situation: Under the assumption that it is possible to find

an explicit parametrization of the eigenvalue variety E0 in variables (s, t, u, v), it should

be absolutely straightforward to determine the subset F ⊂ E0 (strictly) containing the

set P′′ corresponding to p-reps (cf. diagram (2.11)) as

F := E0(W ) ∩ {u2 = 1} ∩ {sptq = 1} (5.26)

134

As is the case for the Whitehead link, we expect the set E0(W )∩{u2 = 1} to decompose

into two (or possibly) more components Ei, corresponding to irreducible and reducible

representations. In contrast to our situation in chapter 2, it might now happen that

each of these components is determined by more than one equation, but we still expect

the defining equations to be polynomials in s and t, as u and v will be fixed as ±1 in

the various components.

We now need to study the intersections of each Ei with the set defined by the filling

condition sptq = 1, thus giving the components of F. Recall that for the Whitehead link

exterior, we trivially obtained the component corresponding to reducible representations

as defined by sp = 1. For the component corresponding to irreducible representations,

we determined the resultant of the defining equation with the polynomial k2 := sptq−1.

We remark that the method of taking the resultant actually also applies in general:

In fact, the condition sp − 1 = 0 in the reducible case can be regarded as the resultant

of the two defining equations: For k1 = sptq − 1 and k′2 = t− 1, we get

Resultant(k1, k′2, t) = det

sp 1

0 −1 1

−1. . .

.... . . 1

0 −1 1

−1 −1

(q+1)×(q+1)

= (−1)q(sp − 1)

(5.27)

This shows that this method should apply also in a more general situation, where there

are possibly more than two components Ei, with each component possibly defined by

more than one equation.

Suppose now in the general situation that a component Ei is given by defining

equations gij = 0, where gij ∈ C[s, t] are polynomials of certain, fixed degrees (not

depending on p and q). We then could nevertheless consider the resultants resp,q,j :=

Res(gij , k2, t), each of which will have a similar structure to the resultant we obtained

in chapter 2. In particular, since the gij are of fixed degrees, we expect that each resp,q,j

135

will be of a degree which is linear in p and q, i.e. of the form xp + yq + z with fixed

integers x, y, z ∈ Z. Since a solution to the system gij = k2 = 0 has to satisfy resp,q,j = 0,

this will determine an upper bound for the number of distinct solutions of the system,

and in most cases it should be possible to in fact determine the exact number.

Because of the relationship between the total minimal Culler-Shalen seminorm

and the number of conjugacy classes of p-reps (cf. chapter 3), we therefore should also

obtain an upper bound for the minimal seminorm. As a result, we expect this bound

to also depend in a linear way on the filling parameters p and q. We boldly formulate

this as:

Conjecture 5.4.7 Let M be a connected, orientable 3-manifold with boundary con-

sisting of two tori, and let M(p/q) be the 3-manifold obtained by filling one of the

boundary tori along the slope determined by the coprime integers p and q. Then the

minimal Culler-Shalen seminorm of M(p/q) is bounded above by an expression of the

form ap+bq+c for some integers a, b, c ∈ Z. These integers only depend on the interval

in which p/q ∈ Q ∪ {∞} lies.

The different intervals arise from the dependence of some boundary slope of M on the

filling slope p/q. With enough care, it should in fact be possible to exactly determine

the minimal seminorm in this way. It seems likely that the upper bounds will also be

attained in the general case.

We remark that even the knowledge of an upper bound for the minimal seminorm

will often be already sufficient to completely determine the Culler-Shalen seminorm,

as we saw for the manifolds Wp/q in section 4.3. For this to work, some values of the

Culler-Shalen norm have to be known for certain slopes (e.g. Seifert filling slopes). We

expect this to be often possible.

It is our intention to test Conjecture 5.4.7 on other 2-component links in the near

future.

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Index

algebraic integer, 79associated

boundary slope, 16ideal point, 16

base orbifold, 88birationally equivalent, 28boundary slope, 13, 81

associated, 16strict, 14, 110strongly detected, 17, 87, 110

character, 11character variety, 2, 11

smooth point, 60Chebyshev polynomials, 45Culler-Shalen norm, 13, 15Culler-Shalen seminorm, 2, 13

minimal, 58minimal total, 59total, 16, 80

curvenon-trivial, 14, 16, 60trivial, 14

Dehn filling, 1hyperbolic, 1

Dehn surgery, see Dehn fillingfundamental theorem, 1

distanceslope, 14

dominating map, 13

eigenvalue resultant, 36, 44normalized, 47

eigenvalue variety, 3, 12detour, 31

elimination ideal, 19essential

lamination, 5

surface, 5

foliationReebless, 4, 112

framing, 117fundamental ball, 15

Groebner basis, 18, 77group

triangle, see triangle groupgroup cohomology, 21

twisted, 21, 61, 68

hyperbolic, 1, 112hyperbolike, 2

ideal pointassociated, 16

knotfigure-8, 3pretzel, 80twist, 3

laminationessential, 5, 112transversely oriented essential, 4, 112

longitude, 12

manifoldhyperbolic, 1, 112Seifert fibered, 88small, 3

mapdominating, 13

mapping class group, 114meridian, 12minimal seminorm, 14model

smooth projective, 14monodromy, 114

142

canonical form, 117cyclically reduced normal form, 116exact decomposition, 117non-canonical form, 117

monomial order, 18multidegree, 18

non-abelian reducible representation, 23non-trivial curve, 16

once-punctured torus bundle, 3, 112, 114, 120characteristic class, 117equivalence, 117framing, 117

standard, 118weak equivalence, 117

orbifold, 88order

monomial, 18

p-rep, 3, 30partial, 32

p-representation, see p-repp-reps, 59parabolic, 3partial p-rep, see p-rep, partialpolynomial

Alexander, 110Chebyshev, see Chebyshev polynomialsnormal form, 19, 26, 68twisted Alexander, see twisted Alexander

polynomialpretzel knot, see knot, pretzel

r-curve, 16R-order tree, 4, 112rep polynomial, 31representation, 2

abelian, 9dihedral, 9, 99discrete faithful, 30equivalent, 9holonomy, 30irreducible, 9lifting, 98

obstruction, 99non-abelian, 9non-abelian reducible, 11, 23, 62, 91PSL2(C), 8reducible, 9SL2(C), 2, 8

representation variety, 9diagonal subvariety, 10triangular subvariety, 10

resp,q, see eigenvalue resultantnresp,q, see eigenvalue resultant, normalizedresultant, 20, 36

Seifert fibered manifold, see manifold, Seifertfibered

seminorm, 14minimal, 14

slope, 1, 13boundary, see boundary slopeexceptional, 1

slope distance, 14small

manifold, 3smooth projective model, 14surface

essential, 5normal, 81

Sylvester matrix, 20

Tk, see Chebyshev polynomialstopology

Zariski, 9torus bundle, see once-punctured torus bundletotal Culler-Shalen seminorm, 16triangle group, 90twist knot, see knot, twisttwisted Alexander polynomial, 62, 64

Uk, see Chebyshev polynomials

varietycharacter, see character varietyeigenvalue, see eigenvalue varietyrepresentation, see representation variety

virtual, 5

143

Z-representable, 125essential lamination, 125essential surface, 5positive first Betti number, 5, 113, 124

virtually, see virtual

W , see Whitehead link exteriorWhitehead link exterior, 2, 24

Dehn filling, 29eigenvalue variety, 27, 33fundamental group, 25integer filling, 120representation variety

diagonal subvariety, 26, 66triangular subvariety, 25, 32

Wlh, see Whitehead link exteriorWp/q, see Whitehead link exterior, Dehn fillingWrh, see Whitehead link exterior

Zariskiclosure, 9tangent space, 22topology, 9