skeins and algebras - louis-hadrien robert | uni.lu · theorem (m - traczyk 1986 [2]) sk n(d2) ∼=...

Skeins and algebras

H.R.Morton

University of Liverpool

Knots Online, May 2020

H.R.Morton Skeins and algebras

Introduction

Algebras are familiar objects.What about skeins?

skein (noun): a loosely coiled length of wool[skane]


Introduction


John Conway had the knack of finding simple memorable terms forhis many innovative ideas.

Among these were the use of the words Skein and Tangle in knottheory.


Algebras

Algebras that lend themselves to pictorial representations

A common relation in algebras

aba = bab

brings to mind the Reidemeister III relation for knot diagrams. Itlies at the heart of diagrammatic representations of many algebras- sometimes in the guise of the Yang-Baxter equations.


Artin’s n-string braid group and its algebraic representation is aprime example of such an interface between algebraic andgeometric representations which is familiar to most of us in theseseminars.The Hecke algebra Hn(z) of type A is the group algebra of thebraid group Bn over Z[z ] with the relation

a2 = 1 + za

for each a = σi , or some rescaled variant of this. It is adeformation of the group algebra of Sn, where z = 0.The quadratic relation, in the form

a − a−1 = z

suggests the diagrammatic picture

− = z ,

which lies at the heart of the Homfly polynomial for knots andlinks.


Motto

Where there is a quadratic relation in an algebra look for someHomfly-based model.

Algebras with suitable relations include

Hecke algebras Hn(z)

Affine Hecke algebras Hn(z)

Double affine Hecke algebras Hn(z , q)

These algebras come with presentations by generators andrelations.We may want to adapt the coordinate ring, for instance tofactorise the quadratic relation as

(a − s)(a + s−1) = 0,

with z = s − s−1, or rescale the roots a bit.


Skeins

The skein side of the picture

This comes from diagrammatic constructions based on framedoriented curves in 3-manifolds, which give some neat models forthe algebras.

These models also highlight certain elements in the algebras thathave memorable pictures.Here is the basic skein framework for these and some furtherinteresting examples of algebras.


Skeins

For a 3-manifold M the (HOMFLYPT) skein Sk(M) is based onoriented framed curves in M up to isotopy.

Take Λ-linear combinations of curves

Factor by three local relations

Quadratic

− = z

Framing

= = v , = = v−1

Unknot

= δ

For compatibility we need δz = v−1 − v in Λ.


Skein algebras

Algebras turn up naturally when M = F × [0, 1] for a surface F .

Stacking copies of F × I defines a product on Sk(F × I ), making itan algebra over Λ.

We can fix n points J ⊂ F and include n arcs from J × {0} toJ × {1}, to give an algebra for each n.

Write Skn(F ) for this algebra - its elements are linear combinationsof n-tangles in (F , J).When n = 0 we just have the skein of closed curves in F × I -write simply as Sk(F ).


The disc

In Skn(F ) the n-braids play an important role - these are tangleswith just n monotonic arcs. If we stick to these, and just imposethe Quadratic relations then the known presentations of surfacebraid groups give a presentation of the resulting algebra.

The simplest case is F = D2, where this leads directly to theHecke algebra Hn(z), having a nice basis of n! elements.

Theorem (M - Traczyk 1986 [2])

Skn(D2) ∼= Hn(z)⊗ Λ


The disc

It is worth noting this element T (n) in Skn(D2) (and so in Hn(z)).

T (n) =

It is clearly central - any tangle can slide down from the top to thebottom.


The disc

Use the skein relations to write it in terms of braids.

= + z .

We finish up with the sum of the Jucys-Murphy elements, Tj .

Theorem (M, 2000 [5])

T (n) = δ + v−1z∑

Tj =

j

∈ Hn.


Symmetric functions of the Jucys-Murphy elements are knownalgebraically to be central in Hn - diagrammatically we can putmore complicated curves around the loop in T (n) to get all thecentral elements.


The annulus

Look now at the skeins based on the annulus A = S1 × I .The case Sk(A) := C with closed curves only, is a commutativealgebra.

Theorem (Turaev 1987 [7])

C is a free polynomial algebra on generators {Am,m ∈ Z− {0}}.

C is very useful in constructing extra skein invariants of knots K byplacing curves in A around a neighbourhood of K to give morecomplicated elements of a skein.

Interpretations of C as a ring of symmetric polynomials incommuting variables {y±1

i } have proved fruitful, [1]. In particular Iwill use the elements Pm ∈ C which correspond to the power sum∑

ymi .


The annulus

In the annulus there are some simple 1-braids, giving elements

Id = , x = , x3 =

in Sk1(A). When we include closed curves in the skein thequadratic relation gives a relation

− = zx = (s − s−1)x

between 1-tangles.


The annulus

If we put the element Pm on the closed curve it has the propertythat

Pm −Pm = (sm − s−m)xm.


The annulus

This leads to the result

Pm

−Pm

= (sm − s−m)∑

Zmi

where

Zi =

i

in Skn(A).


The annulus

The skein Skn(A), restricted to using n-braids, is isomorphic to theaffine Hecke algebra Hn(z). The full skein gives Hn(z) also, withextended coefficients.

Theorem

Skn(A) ∼= Hn(z)⊗ C


The torus

Peter Samuelson and I worked out the structure of the algebraSk(T 2).

Theorem (M, Samuelson 2015 [3])

Sk(T 2) is generated by elements Px for x ∈ Z2 − {0, 0}

represented by simple closed curves in T 2 decorated by Pm.It is not commutative but

[Px,Py] = (sk − s−k)Px+y

where k = det(xy).


The torus

The double affine Hecke algebra Hn of Cherednik is an algebra overZ[s±1, q±1] generated by

{Ti}, 1 ≤ i ≤ n − 1, {Xj}, {Yj}, 1 ≤ j ≤ n

with relations

(Ti + s)(Ti − s−1) = 0

TiTi+1Ti = Ti+1TiTi+1

[Ti ,Tj ] = 0, |i − j | > 1

Xi+1 = TiXiTi ,

Yi+1 = T−1i YiT

−1i

[Ti ,Xj ] = [Ti ,Yj ] = 0, j 6= i , i + 1

[Xi ,Xj ] = [Yi ,Yj ] = 0

X−11 Y2 = Y2X

−11 T−2

1

Y1X1 · · ·Xn = qX1 · · ·XnY1


The torus

This suggests a comparison with Skn(T2) based on n-braids in T 2.

The comparison is partially successful, taking Ti = σ−1i ,

z = s − s−1 and Xi ,Yi to be pure braids where string i moves oncearound the x or y direction in T 2.

However this only works with braids and with q = 1.If we go beyond braids and include closed curves there is a furtherimmediate problem, even for n = 1.


The torus

In T 2 the 1-braids have the form xky l , with yx = xy .Our picture of

x =

in the annulus gives us a view of x in the torus, while y runssimilarly round the y - direction.If we include a closed curve in the torus we realise that

=

since the closed curve at the back can be moved all the way roundthe torus in the y -direction to reappear at the front.


The torus

The equation

− = (s − s−1)

then becomes 0 = (s − s−1)x .In a plan view of the torus we can see this as

− = (s − s−1)


The torus

Samuelson and I have found a fix for both these problems whichintroduces q and avoids this sort of collapse.Put in an extra fixed string ∗ × I ⊂ T 2 × I . Others can passthrough it at the expense of multiplication by q.Viewed from above

* = q * = q *

Then

* − * = (s − s−1) * = (1− q) *


The torus

With braids only we get the relation yx = qxy when n = 1, andmore generally we identify Hn(z , q) with BSkn(T

2, ∗) using braidsonly, along with the fixed string ∗. (In a recent arXiv preprint [4]).

We then have a homomorphism

ϕn : Hn(z , q) → Skn(T2, ∗).

So long as q 6= 1 we avoid the immediate danger of the full skeinSkn(T

2, ∗) collapsing.Samuelson and I show that ϕn is surjective. It would be good tobe certain that it is also injective when q 6= 1.


The torus

With braids only we get the relation yx = qxy when n = 1, andmore generally we identify Hn(z , q) with BSkn(T

2, ∗) using braidsonly, along with the fixed string ∗. (In a recent arXiv preprint [4]).

We then have a homomorphism

ϕn : Hn(z , q) → Skn(T2, ∗).

So long as q 6= 1 we avoid the immediate danger of the full skeinSkn(T

2, ∗) collapsing.Samuelson and I show that ϕn is surjective. It would be good tobe certain that it is also injective when q 6= 1.

We get nice representations of elements of Hn such as∑

Xmi and∑

Ymi by quite simple elements in the full skein. These are to a

large extent independent of n.


The torus

Using a plan view of the torus, with the braid points and fixedstring where indicated, we have diagrams of the braids

Xi = * , Yi = *

Our result from the annulus gives us

*

Pm

− *Pm = (sm − s−m)

∑

i

Xmi


The torus

This leads to

(sm − s−m)∑

i

Xmi = (1− qm) *

Pm

This closed curve element of the skein fits nicely with the results ofSchiffman and Vasserot [6] relating the elliptic Hall algebra to alimit of quotients of the algebras Hn.


The elliptic Hall algebra

The elliptic Hall algebra has generators ux for x 6= 0 ∈ Z2, and SV

look at images of them in quotients of Hn.

Key elements in their model are the power sums

Xm1 + · · ·+ Xm

n ,Ym1 + · · ·+ Ym

n

and their images under automorphisms of Hn.

We can use closed curves in the full skein to set up elements Wx

that are essentially independent of n, and correspond quite neatlywith ux in the quotients of each Skn(T

2, ∗).


Construction of elements Wx

Fix a disc D in T 2 which includes the braid points and the basepoint. Any oriented embedded curve in the complement of D2 isdetermined up to isotopy by a primitive element y ∈ Z

2,representing the homology class of the curve.

For each primitive y define an element Wy of the skein Skn(T2, ∗)

by the oriented curve corresponding to y, along with vertical braidstrings and base string.


Construction of elements Wx

For any other non-zero x ∈ Z2 write x = my with m > 0 and y

primitive, and define Wx to be Wy with the closed curve decoratedby the element Pm.

These give elements of Skn(T2, ∗) for each n. Our example above

is

Wm,0 = *

Pm

In the correspondence with Hn we think of comparing Wx and(sm − s−m)ux.


The elements Wx really belong in the skein Sk(T 2 − D2). If we fillin D2 they become the elements Px ∈ Sk(T 2). They satisfy somebut not all of the commutation relations from Sk(T 2).

The skein Sk(T 2 − D2) is much larger than Sk(T 2). A part of itcan be mapped to a part of the elliptic Hall algebra, using the limitresult in [6], with Wx going to a multiple of ux. This depends onthe homomorphisms ϕn : Hn → Skn(T

2, ∗) turning out to beisomorphisms.


References I

Liverpool Knot Theory publications link.https://www.liverpool.ac.uk/˜su14/knotprints.html

[1] H. R. Morton and P. M. G. Manchon.Geometrical relations and plethysms in the Homfly skein of theannulus.J. Lond. Math. Soc. (2), 78(2):305–328, 2008.

[2] H. R. Morton and P. Traczyk.Knots and algebras.In ‘Contribuciones Matematicas en homenaje al profesor D.Antonio Plans Sanz de Bremond,’ ed. E. Martin-Peinador andA. Rodez Usan, University of Zaragoza, 201-220. 1990.


References II

[3] Hugh Morton and Peter Samuelson.The HOMFLYPT skein algebra of the torus and the ellipticHall algebra.Duke Math. J., 166(5):801–854, 2017.

[4] Hugh Morton and Peter Samuelson.DAHAs and skein theory.arXiv 1909.11247, September 2019.

[5] Hugh R. Morton.Skein theory and the Murphy operators.J. Knot Theory Ramifications, 11(4):475–492, 2002.Knots 2000 Korea, Vol. 2 (Yongpyong).


References III

[6] O. Schiffmann and E. Vasserot.The elliptic Hall algebra, Cherednik Hecke algebras andMacdonald polynomials.Compos. Math., 147(1):188–234, 2011.

[7] V. G. Turaev.The Conway and Kauffman modules of a solid torus.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.(LOMI), 167(Issled. Topol. 6):79–89, 190, 1988.


skeins and algebras - louis-hadrien robert | uni.lu · theorem (m - traczyk 1986 [2]) sk n(d2) ∼=...

Documents