garside structure and dehornoy ordering of braid groups...

Garside structure and Dehornoy ordering of braidgroups for topologist (mini-course I)

Tetsuya Ito

Combinatorial Link Homology Theories, Braids, and Contact GeometryAug, 2014

Tetsuya Ito Braid calculus Sep , 2014 1 / 98

Contents

Introduction

Part I: Garside theory of braid groups

I-1: Toy model: Garside structure on Z2

I-2: Classical Garside structure

I-3: Dual Garside structure

I-4: Application to topology (1): Nielsen-Thurston classification

I-5: Application to topology (2): Curve diagram and linearrepresentation

Introduction

Braid group

The n-strand braid group

⟨σ1, . . . , σn−1

σiσjσi = σiσjσi , |i − j | = 1σiσj = σjσi , |i − j | > 1

1 2 i i+ 1 n1 n

An element of Bn is represented by n-strings (braid) in C× [0, 1].We have a natural map π : Bn → Sn, and the pure braid group Pn is thekernel of π.

Braid group in topology (I) relation to knot theory

Alexander-Markov Theorem

Braids/conjugation,stabilization) 1:1←→ Oriented links in S3

Braid group in topology (I) relation to knot theory

Transverse Markov Theorem (Orevkov-Shevchishin, Wrinkle ’02)

Braids/conjugation, positive stabilization)1 : 1

Transverse links in standard contact S3

Braid group in topology (II) relation to MCG

Dn = z ∈ C | |z | ≤ n + 1 − 1, . . . , n: n-punctured disc

Bn∼= MCG (Dn)

= Mapping class group of Dn

= f : DnHomeo−→ Dn | f |∂Dn = id/Isotopy

σi ↔ Half Dehn-twist swapping i and (i + 1).

Dehn-twist

Braid group in topology (II) relation to MCG

Dn = z ∈ C | |z | ≤ n + 1 − 1, . . . , n: n-punctured disc

Bn∼= MCG (Dn)

= Mapping class group of Dn

= f : DnHomeo−→ Dn | f |∂Dn = id/Isotopy

σi ↔ Half Dehn-twist swapping i and (i + 1).

Dehn-twist

Braid group in topology (III) configuration space

The ordered/unordered configuration space of n-points in C:

Cn(C) = (z1, . . . , zn) ∈ Cn | zi = zj if i = j, UCn(C) = Cn(C)/Sn

Based loops in UCn(C) are naturally regarded as braids soΩUCn(C) = Space of braidsπ1(Cn(C)) = Pn, π1(UCn(C)) = Bn.

A natural projection

p : Cn(C)→ Cn−1(C), p(z1, . . . , zn) 7→ (z1, . . . , zn−1)

is a fibration with fiber C− (n − 1) points, with section

s : Cn−1(C)→ Cn(C), s(z1, . . . , zn−1,max|zi |+ 1).

This shows

Theorem (Atrin ’47, Fox-Neuwirth ’62, Fadell-Neuwirth ’62)

1. Cn(C) = K (Pn, 1), UCn(C) = K (Bn, 1)

2. The cohomological dimension of Bn and Pn are finite, and both Bn

and Pn are torsion-free.

3. Pn = Fn−1 ⋊ Pn−1 = (Fn−1 ⋊ (Fn−2 ⋊ (Fn−3 · · · (F2 ⋊ F1)) · · · ).

A natural projection

p : Cn(C)→ Cn−1(C), p(z1, . . . , zn) 7→ (z1, . . . , zn−1)

is a fibration with fiber C− (n − 1) points, with section

s : Cn−1(C)→ Cn(C), s(z1, . . . , zn−1,max|zi |+ 1).

This shows

Theorem (Atrin ’47, Fox-Neuwirth ’62, Fadell-Neuwirth ’62)

1. Cn(C) = K (Pn, 1), UCn(C) = K (Bn, 1)

2. The cohomological dimension of Bn and Pn are finite, and both Bn

and Pn are torsion-free.

3. Pn = Fn−1 ⋊ Pn−1 = (Fn−1 ⋊ (Fn−2 ⋊ (Fn−3 · · · (F2 ⋊ F1)) · · · ).

Braid group in topology (IV) Hyperplane arrangement

Cn(C) is regarded as the complement of an hyperplane arrangement calledthe braid arrangement:For 1 ≤ i < j ≤ n, let

Hi ,j = Ker(zi − zj) ⊂ Cn, A = Hi ,j1≤i<j≤n

ThenCn(C) = M(A) = Cn −

∪1≤i<j≤n

Hi ,j ,

1. Reflections with respect to Hi ,j ’s forms the symmetric group Sn. Close and natural connection between root systems, Coxeter groups

and Artin groups. Source of combinatorics of braids.

2. A well-known method to construct cellular decomposition of M(A)(Salvetti complex) gives a presentation of Bn.

Braid group in topology (IV) Hyperplane arrangement

Cn(C) is regarded as the complement of an hyperplane arrangement calledthe braid arrangement:For 1 ≤ i < j ≤ n, let

Hi ,j = Ker(zi − zj) ⊂ Cn, A = Hi ,j1≤i<j≤n

ThenCn(C) = M(A) = Cn −

∪1≤i<j≤n

Hi ,j ,

1. Reflections with respect to Hi ,j ’s forms the symmetric group Sn. Close and natural connection between root systems, Coxeter groups

and Artin groups. Source of combinatorics of braids.

2. A well-known method to construct cellular decomposition of M(A)(Salvetti complex) gives a presentation of Bn.

Part I: Garside theory for braid groups

Word and conjugacy problem

Word/Conjugacy Problem

For given braids α, β (as a word over σ±11 , . . . , σ±1

n−1) Determine whether α = β or not.

Determine whether α and β are conjugate or not.(and, find γ such that γβγ−1 = α.)

Since group is suited for computations (encoding), our ultimate goal is:

Algebraic link Problem

For two links (represented as closed braids),

Determine whether they are the same or not

Determine basic properties (prime/split/satellite/hyperbolic,etc...)

Word/conjugacy problem is the first step towards this problem.

Word and conjugacy problem

Word/Conjugacy Problem

For given braids α, β (as a word over σ±11 , . . . , σ±1

n−1) Determine whether α = β or not.

Determine whether α and β are conjugate or not.(and, find γ such that γβγ−1 = α.)

Since group is suited for computations (encoding), our ultimate goal is:

Algebraic link Problem

For two links (represented as closed braids),

Determine whether they are the same or not

Determine basic properties (prime/split/satellite/hyperbolic,etc...)

Word/conjugacy problem is the first step towards this problem.

What is Garside theory ?Garside theory (Garside structure) is a machinery of:

1. Producing the normal form of a group. Easy to calculate (and suited for computor) Idea and its meaning sounds natural.

2. Giving several nice structures of the group (automatic, lattice...)

3. Allowing us to solve other problems (conjugacy, extracting roots,etc...)

In particular, for the case of braid groups:

Garside structure yields “the best” normal form – it reflects

Dynamics (Nielsen-Thurston classification)

Topology (infinite cyclic coverings)

Algebra (quantum/homological representation)

Dehornoy’s ordering

What is Garside theory ?Garside theory (Garside structure) is a machinery of:

1. Producing the normal form of a group. Easy to calculate (and suited for computor) Idea and its meaning sounds natural.

2. Giving several nice structures of the group (automatic, lattice...)

3. Allowing us to solve other problems (conjugacy, extracting roots,etc...)

In particular, for the case of braid groups:

Garside structure yields “the best” normal form – it reflects

Dynamics (Nielsen-Thurston classification)

Topology (infinite cyclic coverings)

Algebra (quantum/homological representation)

Dehornoy’s ordering

I-1: Toy model: Garside structure on Z2

Toy model: Garside structure on Z2

G = Z2 = ⟨x , y⟩: Free abelian group of rank twoP = xayb | a, b ≥ 0: set of “positive” elements∆ = xy = yx : Garside element

Key features:

P is a submonoid: α, β ∈ P ⇒ αβ ∈ P.

For any α ∈ G , ∆nz ∈ P for sufficiently large n.

For α = xayb, β = xa′yb

′ ∈ G , define

α ≼ β ⇐⇒ a ≤ a′ and b ≤ b′

⇐⇒ α−1β ∈ P.

Then x , y ≼ ∆.

[1,∆]Def= β ∈ G | 1 ≼ β ≼ ∆ = 1, x , y ,∆.

Toy model: Garside structure on Z2

G = Z2 = ⟨x , y⟩: Free abelian group of rank twoP = xayb | a, b ≥ 0: set of “positive” elements∆ = xy = yx : Garside element

Key features:

P is a submonoid: α, β ∈ P ⇒ αβ ∈ P.

For any α ∈ G , ∆nz ∈ P for sufficiently large n.

For α = xayb, β = xa′yb

′ ∈ G , define

α ≼ β ⇐⇒ a ≤ a′ and b ≤ b′

⇐⇒ α−1β ∈ P.

Then x , y ≼ ∆.

[1,∆]Def= β ∈ G | 1 ≼ β ≼ ∆ = 1, x , y ,∆.

Normal form for Z2

Definition

For β ∈ G , the normal form of β is a word over x , y ,∆±1

N(β) = ∆ps1s2 · · · sr (p ∈ Z, si ∈ x , y ,∆)

such that

1. ∆−pβ ∈ P.

2. si is the ≼-largest element among x , y ,∆ satisfying

si ≼ (s−1i−1 · · · s

−11 ∆−p)β

(So normal form of β = xayb is actually written as:

N(β) =

∆ayb−a b ≥ a

∆bxa−b a ≥ b

Normal form for Z2

Definition

For β ∈ G , the normal form of β is a word over x , y ,∆±1

N(β) = ∆ps1s2 · · · sr (p ∈ Z, si ∈ x , y ,∆)

such that

1. ∆−pβ ∈ P.

2. si is the ≼-largest element among x , y ,∆ satisfying

si ≼ (s−1i−1 · · · s

−11 ∆−p)β

(So normal form of β = xayb is actually written as:

N(β) =

∆ayb−a b ≥ a

∆bxa−b a ≥ b

What is the meaning of normal form ?

Normal form = path in the Cayley graph which approaches the destinationin the “fastest” way at any intermediate time.

Q: How to go back to home from university ?

University

We are tired, so we want to go back to home as early as possible...

Normal form = path in the Cayley graph which approaches destination inthe “fastest” way at any intermediate time.

University

yyxyyx

This path is not effective (geodesic) – we can do several short-cuts.

Normal form = path in the Cayley graph which approaches the destinationin the “fastest” way at any intermediate time.

University

These paths are both geodesic (so the total arrival time is the same) but...

Normal form = path in the Cayley graph which approaches destination inthe “fastest” way at any intermediate time.

University

Normal form

After 2minutes, normal form path lies closer than other path.

How to computing normal forms ?

Strategy to get normal form

1. By considering ∆nβ for sufficiently large n, we assume β ∈ P .

2. Starting at the final destination, we do: let us look sub-path si si+1: check whether this sub-path is “nice” of

not (whether this sub-path is a normal form or not) If this sub-path is not nice (i.e. we are going by a roundabout route)

replace this sub-path si si+1 with better one (tighten locally).

Crucial fact

By resolving local roundabouts, we will eventually get globally nice path,the normal form.

(cf. Length of geodesic connecting two point x , y in Riemannian manifold= distance of x and y)

How to computing normal forms ?

1. By considering ∆nβ for sufficiently large n, we assume β ∈ P .

2. Starting at the final destination, we do: let us look sub-path si si+1: check whether this sub-path is “nice” of

not (whether this sub-path is a normal form or not) If this sub-path is not nice (i.e. we are going by a roundabout route)

replace this sub-path si si+1 with better one (tighten locally).

Crucial fact

By resolving local roundabouts, we will eventually get globally nice path,the normal form.

(cf. Length of geodesic connecting two point x , y in Riemannian manifold= distance of x and y)

Computing normal forms: example

The best hoi e

of the rst path

The rst letter of

the normal form

The best hoi e

of the se ond path

The se ond letter of

the normal form

The normal form is:

Computing normal forms: conclusion

How fast can we compute the normal form ?Previous argument says:

Conclusion

For β ∈ G of length ℓ (as a word over x , y ,∆), after performingℓ(ℓ−1)

2 = O(ℓ2) times of “local tightening” (replacing local roundaboutroute with the best one), we are able to get a normal form of β.

Moreover, note that in the process of local tightening, we just need to lookat the path of length two. This says

Conclusion’

To compute normal form, we only need finite data (of which path isbetter).

Computing normal forms: conclusion

How fast can we compute the normal form ?Previous argument says:

Conclusion

For β ∈ G of length ℓ (as a word over x , y ,∆), after performingℓ(ℓ−1)

2 = O(ℓ2) times of “local tightening” (replacing local roundaboutroute with the best one), we are able to get a normal form of β.

Moreover, note that in the process of local tightening, we just need to lookat the path of length two. This says

Conclusion’

To compute normal form, we only need finite data (of which path isbetter).

I-2: Classical Garside structure

General idea of Garside structure

We want to generalize idea and method for “toy model” for more generaland complicated group G – what we need ?In toy model, we have used:

1. Submonoid P consisting of “positive elements”:P consists of positive words over certain generating sets x , y , . . . , of G .

The notion of positive elements yields a subword ordering ≼:

α ≼ βDef⇐⇒ α−1β ∈ P.

2. Special element ∆: For any β ∈ G , ∆nβ ∈ P for sufficiently large n. x , y , . . . ≼ ∆.

By giving “good” ∆ and P , one can generalize the toy model idea.

General idea of Garside structure

We want to generalize idea and method for “toy model” for more generaland complicated group G – what we need ?In toy model, we have used:

1. Submonoid P consisting of “positive elements”:P consists of positive words over certain generating sets x , y , . . . , of G .

The notion of positive elements yields a subword ordering ≼:

α ≼ βDef⇐⇒ α−1β ∈ P.

2. Special element ∆: For any β ∈ G , ∆nβ ∈ P for sufficiently large n. x , y , . . . ≼ ∆.

By giving “good” ∆ and P , one can generalize the toy model idea.

The classical Garside structure of braid

B+n = Product of σ1, . . . , σn−1 : Positive braid monoid

∆ = (σ1σ2 · · ·σn−1)(σ1σ2 · · ·σn−2) · · · (σ1σ2)(σ1) : Garside element

Definition-Proposition

Define the relation ≼ of Bn by x ≼ y ⇐⇒ x−1y ∈ B+n . Then ≼ is a

lattice ordering:

≼ admits the greatest common divisor

x ∧ y = max≼z ∈ Bn | z ≼ x , y

≼ admits the least common multiple

x ∨ y = min≼z ∈ Bn | x , y ≼ z

σ1, σ2, . . . , σn−1 ≼ ∆.

The classical Garside structure of braid

B+n = Product of σ1, . . . , σn−1 : Positive braid monoid

∆ = (σ1σ2 · · ·σn−1)(σ1σ2 · · ·σn−2) · · · (σ1σ2)(σ1) : Garside element

Define the relation ≼ of Bn by x ≼ y ⇐⇒ x−1y ∈ B+n . Then ≼ is a

lattice ordering:

x ∧ y = max≼z ∈ Bn | z ≼ x , y

x ∨ y = min≼z ∈ Bn | x , y ≼ z

σ1, σ2, . . . , σn−1 ≼ ∆.

Why we choose such ∆ and B+n ?

We want to define the normal form

N(β) = ∆ps1 · · · sr

as we have done in the case Z2 (toy model): So we first need

∆−pβ ∈ B+n

and s1 should be:

the ≼ -maximal element satisfying s1 ≼ ∆−pβ (∈ B+n )

⇒ We need to know such ≼-maximal element always exists⇒ Lattice structure naturally appear.

Simple braids

≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ32︸︷︷︸

Positive braids

∆ = σ1 ∨ σ2 ∨ · · · ∨ σn−1.

definition

A simple braid is a braid that satisfies 1 ≼ x ≼ ∆.

Note: B+n = Product of σ1, . . . , σn−1

= Product of simple braids

Proposition

[1,∆]Def= simple braids 1:1←→ Sn

(so simple braids are often called premutation braids)

Simple braids

≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ32︸︷︷︸

Positive braids

∆ = σ1 ∨ σ2 ∨ · · · ∨ σn−1.

definition

A simple braid is a braid that satisfies 1 ≼ x ≼ ∆.

Note: B+n = Product of σ1, . . . , σn−1

= Product of simple braids

Proposition

[1,∆]Def= simple braids 1:1←→ Sn

(so simple braids are often called premutation braids)

Example: B3 case

∆ = (σ1σ2)σ1 = σ2σ1σ2, so

[1,∆] = 1, σ1, σ2, σ1σ2, σ2σ1,∆

Simple braids: each strand positively crosses with other strands at mostonce.

Normal form

Theorem-Definition (Garside, Elrifai-Morton, Thurston)

A braid β ∈ Bn admits the normal form

N(β) = ∆px1x2 · · · xr (p ∈ Z, xi ∈ [1,∆])

1. ∆−pβ ∈ B+n .

2. xi = ∆ ∧ (x−1i−1 · · · x

−11 ∆−pβ).

By absorbing first few ∆ terms in x1, . . ., N(β) is uniquely written as

N(β) = ∆px1x2 · · · xr (p ∈ Z, xi = ∆).

We define the infimum, supremum of β by

inf(β) = p, sup(β) = p + r .

How to compute normal form ?

As in the toy model case, a word is a normal form if and only if it is locallya normal form:

Theorem (Elrifai-Morton, Thurston)

A wordN ′(β) = ∆px1x2 · · · xr (p ∈ Z, xi ∈ [1,∆])

is a normal form if and only if

(xixi+1) ∧∆ = xi for all i

(i.e., xixi+1 is also a normal form)

The strategy for computing normal form applies to the braid group case:

1. Express β as a word of the form

β = ∆px1 · · · xr (p ∈ Z, xi ∈ [1,∆])

∆2 = (σ1σ2 · · ·σn−1)n is the full-twist braid (as an element of

MCG (Dn), it is the Dehn twist along ∂Dn), which is a generator of thecenter of Bn, so

· · ·σ−1i · · · = · · ·∆

−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1

i )︸︷︷︸Positive braid

· · ·

2. Apply local tightening repeatedly: for i = r , . . . , 1 rewrite eachsub-path xixi+1 so that it is a normal form

xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆

β = ∆px1 · · · xr (p ∈ Z, xi ∈ [1,∆])

· · ·σ−1i · · · = · · ·∆

−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1

· · ·

xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆

β = ∆px1 · · · xr (p ∈ Z, xi ∈ [1,∆])

· · ·σ−1i · · · = · · ·∆

−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1

· · ·

xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆

Simple example

Let us compute the normal form of a 3-braid β = (σ−12 )(σ1σ2)(σ2)(σ1σ2).

1. Rewriting β as the form ∆p (positive braids):

β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)

2. Apply local tightenings for

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

to get normal forms

Simple example

β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

to get normal forms

Simple example

β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

to get normal forms

Simple example: local tightening

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

Hence β = ∆−1β′ = ∆−1∆∆σ2 and its normal form is

N(β) = (∆)(σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

N(β) = (∆)(σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

N(β) = (∆)(σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

N(β) = (∆)(σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

N(β) = (∆)(σ2)

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

(σ2)(σ1σ2) ∧∆ = ∆, so

β′ = (σ1σ2)(σ1σ2)(∆).

(σ1σ2)(∆) ∧∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so

β′ = (σ1σ2)(∆)(σ2σ1)

(σ1σ2)(∆) ∧∆ = ∆, so

β′ = ∆(σ2σ1)(σ2σ1)

(σ1σ2)(σ1σ2) ∧∆ = ∆, and (σ1σ2)(σ1σ2) = (∆)(σ2), so

β′ = ∆∆σ2.

N(β) = (∆)(σ2)

Meaning of normal form condition

What is the meaning of condition (xixi+1) ∧∆ = xi ?

Proposition

For x ∈ [1,∆], define the starting set S(x) by

S(x) = σi | x = σi · (positive braid) (i .e. σi ≼ x)

and the finishing set F (x) by

F (x) = σi | x = (positive braid) · σi

Then for simple braids x and y ,

xy ∧∆ = x ⇐⇒ F (x) ⊃ S(y)

Meaning of normal form condition

The situation F (x) ⊃ S(y) prevents to absorb crossings in y into x :

(Recall that:simple braid ⇐⇒ each pair of strand crosses at most by once

For es to have se ond rossings

between two strands

Geodesic property

x−1∆ and x∆ = ∆x∆−1 are simple if x is simple.

Rewrite a normal form N(β) = ∆px1 · · · xr as

W (β) =

∆px1 · · · xr (p > 0)

(∆−1x1)∆p+1

(∆−1x2)∆p+2 · · · (∆−1x−p)x−p+1 · · · xr

(p < 0, p + r > 0)

(∆−1x1)∆p+1 · · · (∆−1xr )

∆p+r∆−p−r (p + r < 0)

Theorem (Charney)

W (β) is a geodesic word. So the length of β (with respect to simplebraids [1,∆] is

ℓ[1,∆](β) = maxsup(β), 0 −mininf(β), 0.

Geodesic property

x−1∆ and x∆ = ∆x∆−1 are simple if x is simple.

Rewrite a normal form N(β) = ∆px1 · · · xr as

W (β) =

∆px1 · · · xr (p > 0)

(∆−1x1)∆p+1

(∆−1x2)∆p+2 · · · (∆−1x−p)x−p+1 · · · xr

(p < 0, p + r > 0)

(∆−1x1)∆p+1 · · · (∆−1xr )

∆p+r∆−p−r (p + r < 0)

Theorem (Charney)

W (β) is a geodesic word. So the length of β (with respect to simplebraids [1,∆] is

ℓ[1,∆](β) = maxsup(β), 0 −mininf(β), 0.

Normal form produces automatic structure

The characterizing property of normal form is “local”(we only need to see consecutive factor xixi+1)

Theorem (Thurston, Charney, Dehornoy)

The normal forms of Bn provides a geodesic automatic structure. Inparticular,

Set of normal forms 1:1←→ Path of certain graph (automata)

Example: Automata for B3

Normal formN(β) = ∆−1∆−1(σ2σ1)(σ1σ2)(σ2σ1)

Conjugacy problem (I)

Using normal form technique, we can solve the conjugacy (search)problem.

Basic strategy

For given α ∈ Bn, try to determine the set of “simplest” normal formsamong its conjugacy class, called ... summit set.

S(α) =

β is conjugate to α, with the “simplest”N(β)+“Additional requirements”

S(α) = S(α′) ⇐⇒ α and β are conjugate

Conjugacy problem (II)

By cycling and decycling operation, we may find simpler normal formamong the conjugacy class of given braid β:

: simpler normal form

de y ling

y ling

It may happen p′ > p or r ′ < r

Conjugacy problem (II)

Theorem (Garside, ElRifai-Morton, Gebhardt, Gonzalez-Meneses)

Let α ∈ Bn.

1. By applying cycling and decylings finitely many times, we can findone element in S(α).

2. Staring from one element β ∈ S(α), by repeatedly computing theconjugate of β by simple elements, we can find all elements of S(α):

In particular, we have an algorithm to solve the conjugacy decision andproblem (determine α ∼conj α

′) and the conjugacy search problem (find βsuch that α = β−1α′β).

Conjugacy problem (II) example of “... (summit) set”The super summit set

SS(α) =

β is conjugate to α withmaximal inf,minimum sup

The ultra summit set

US(α) = β ∈ SS(α) | closed under cycling operation

Conjugacy problem (III)

Using idea of summit set, we can solve the conjugacy problem (but in timeO(e length), in general):

Computing a normal form is easy (done in polynomial time).

Starting from α, finding one element of S(α) is (conjecturally) donein polynomial time.

Size of S(α) might be quite huge – the size of S(α) might beO(e length) (So computing whole S(α) might require exponentialtimes...)

Problem

Find polynomial time algorithm for conjugacy problem of braids.

Problem

Understand the structure of summit sets.

Conjugacy problem (III)

Using idea of summit set, we can solve the conjugacy problem (but in timeO(e length), in general):

Computing a normal form is easy (done in polynomial time).

Starting from α, finding one element of S(α) is (conjecturally) donein polynomial time.

Size of S(α) might be quite huge – the size of S(α) might beO(e length) (So computing whole S(α) might require exponentialtimes...)

Problem

Find polynomial time algorithm for conjugacy problem of braids.

Problem

Understand the structure of summit sets.

I-3: Dual Garside structure

Dual Garside structureThe braid group has another Garside structure called dual Garsidestructure, by consdiering different P (the set of positive elements) and δ(Garside element)

Definition

For 1 ≤ i < j ≤ n, let

ai ,j = (σi+1 · · ·σj−2σj−1)−1σi (σi+1 · · ·σj−2σj−1)

The generating set Σ∗ = ai ,j1≤i<j≤n is called a dual Garside generator(Birman-Ko-Lee generator or band generator).

1 i j i jn

twisted band

Dual Garside structure

B+∗n = Product of positive ai ,j : Dual positive monoid

δ = σ1σ2 · · ·σn−1 = a1,2a2,3 · · · an−1,n : Dual Garside element

Define the relation ≼∗ of Bn by x ≼∗ y ⇐⇒ x−1y ∈ B+∗n . Then ≼ is a

lattice ordering:

x ∧∗ y = max≼∗z ∈ Bn | z ≼∗ x , y

x ∨∗ y = min≼∗z ∈ Bn | x , y ≼∗ z

ai ,j ≼∗ δ for all 1 ≤ i < j ≤ n.

B+∗n = Product of positive ai ,j : Dual positive monoid

δ = σ1σ2 · · ·σn−1 = a1,2a2,3 · · · an−1,n : Dual Garside element

Define the relation ≼∗ of Bn by x ≼∗ y ⇐⇒ x−1y ∈ B+∗n . Then ≼ is a

lattice ordering:

x ∧∗ y = max≼∗z ∈ Bn | z ≼∗ x , y

x ∨∗ y = min≼∗z ∈ Bn | x , y ≼∗ z

ai ,j ≼∗ δ for all 1 ≤ i < j ≤ n.

Definition

A dual simple braid is a braid that satisfies 1 ≼∗ x ≼∗ δ.

[1, δ] = β ∈ Bn | 1 ≼∗ β ≼∗ δ = Dual simple braids

Theorem-Definition (Birman-Ko-Lee)

A braid β ∈ Bn admits the unique the normal form ( dual Garside normalform)

N∗(β) = δpd1d2 · · · dr (p ∈ Z, xi ∈ [1, δ])

which is characterized by

1. p = minn ∈ Z | δnβ ∈ B+∗n

2. xi = δ ∧∗ (d−1i−1 · · · d

−11 δ−pβ).

We define the dual supremum, dual infimum of β by

sup ∗(β) = p + r , inf ∗(β) = p

A parallel argument applies for the dual Garside structure:

Theorem (Birman-Ko-Lee)

The dual normal form provides an automatic structure.

Theorem (Birman-Ko-Lee)

An appropriate modification of dual normal form provides a geodesic wordwith respect to the length ℓ[1,δ]. In particular,

ℓ[1,δ](β) = maxsup ∗(β), 0 −mininf ∗(β), 0.

By the similar method, one can use dual normal form to solve theconjugacy problem.

Example: 3-braid case

δ = a1,2a2,3 = a2,3a1,3 = a1,3a1,2, so

[1, δ] = 1, a1,2, a2,3, a1,3, δ

Recall that: Classical simple elements [1,∆]1:1↔ Permutations Sn

What is the (combinatorial) meaning of dual simple elements ?

To treat dual Garside elements, it is convenient to n-punctured disc Dn

with circular symmetry:

Example: 3-braid case

δ = a1,2a2,3 = a2,3a1,3 = a1,3a1,2, so

[1, δ] = 1, a1,2, a2,3, a1,3, δ

Recall that: Classical simple elements [1,∆]1:1↔ Permutations Sn

What is the (combinatorial) meaning of dual simple elements ?

To treat dual Garside elements, it is convenient to n-punctured disc Dn

with circular symmetry:

A geometric understanding of dual simple elementsLet us identify Bn with MCG (Dn). Then,

Proposition (Bessis)

Set of convex polygons in Dn1:1←→ [1, δ]

(Convex polygons is understood as non-crossing partition of n-points)

A geometric understanding of the normal form conditionLike classical Garside case, we have geometric useful interpretation of thenormal form condition δ ∧∗ (xy) = x .

Proposition

For x , y ∈ [1, δ],

δ ∧∗ (xy) = x ⇐⇒ Corresponding convex polygons x are “linked” to y

Linked Not Linked

Open problem

Are there other “Garside structures” (i.e. the submonoid P and element ∆which allows us to develop a machinery for normal forms) for Bn ?

Open problem

Clarify the meaning of the word “dual”:Currently, we use the name “dual” Garside structure because of numericalcorrespondence of several data of the Garside structures (numbers ofatoms, simple elements, ...) and there is no theoretical “duality” at all !

I-3: Application to topology (1)Nielsen-Thurston classification

Nielsen-Thurston theory

According to the dynamics of Bn∼= MCG (Dn), a braid β viewed as a

homeomorphism, β : Dn → Dn is classified into one of the following threetypes: Periodic, reducible, pseudo-Anosov

1: Periodic

βn = ∆2m for some n,m ∈ Z(i.e., Powers of β = Dehn twists along ∂Dn)

2: Reducible

β(C ) = C for some essential simple closed curves C ⊂ Dn

(A simple curve is essential ⇐⇒ C encloses more than one punctures andis not isotopic to ∂Dn)

1: Periodic

2: Reducible

1: Periodic

2: Reducible

3: Pseudo-Anosovβ is a pseudo-Anosov homomorphism (locally, there are β is λ-expandingin one direction and λ-shrinking in transverse direction for some λ > 1(This λ is called the dilatation)

Knowing the Nielsen-Thurston type is important in dynamics, topology(and algebraic properties like centralizers), so

Problem

How to determine the Nielsen-Thurston type of β ?

Train-track method (graph encoding of surface automorphisms) provides asolution of this problem (Bestvina-Handel algorithm).

Now, Garside theory (normal form) provides alternative solution !

Knowing the Nielsen-Thurston type is important in dynamics, topology(and algebraic properties like centralizers), so

Problem

How to determine the Nielsen-Thurston type of β ?

Train-track method (graph encoding of surface automorphisms) provides asolution of this problem (Bestvina-Handel algorithm).

Now, Garside theory (normal form) provides alternative solution !

Nielsen-Thurston type via Garside theoryRecognizing a periodic braid is easy:

Theorem (Eilenberg, Kerekjarto)

A periodic n-braid is conjugate to (σ1σ2 · · ·σn−1)m or (σ1σ2 · · ·σn−1σ1)

m.In particular, if β is periodic, then βn or β(n−1) is a power of ∆2.

The problem is how to recognize a reducible braid.

Why recognizing reducible braid is not so easy ? Because, β may preservevery,very,very complicated “simple” (so it is not simple – rather complex!!!) closed curve.

Assume β is reducible. If N(β) is simple among its conjugacy class, then βpreserves “simple” (not complicated, near “standard”) simple closedcurves.

Simple normal form⇐⇒ Preserving “simple” simple closed curve

Nielsen-Thurston type via Garside theoryRecognizing a periodic braid is easy:

Theorem (Eilenberg, Kerekjarto)

A periodic n-braid is conjugate to (σ1σ2 · · ·σn−1)m or (σ1σ2 · · ·σn−1σ1)

m.In particular, if β is periodic, then βn or β(n−1) is a power of ∆2.

The problem is how to recognize a reducible braid.

Why recognizing reducible braid is not so easy ? Because, β may preservevery,very,very complicated “simple” (so it is not simple – rather complex!!!) closed curve.

Assume β is reducible. If N(β) is simple among its conjugacy class, then βpreserves “simple” (not complicated, near “standard”) simple closedcurves.

Simple normal form⇐⇒ Preserving “simple” simple closed curve

Easy, but informative observation

Observation

For simple braids x , y , if xy is a normal form preserving “standard” roundcurve patterns, then x and y also preserves such a curve pattern.

Nielsen-Thurston type via Garside theory

Theorem (Barnadete-Nitecki-Gutierrez ’95)

If β is reducible, then there exists α ∈ US(β) ⊂ SS(β) such that αpreserves standard a round curve. Thus by computing US(β) or SS(β), wecan determine whether β is reducible or not.

Proof: If β is reducible, by conjugating, β preserves standard round curve.By previous observation, (de)cycling of β has the same property.

Drawback

The theorem says at least one element in US(β) is very nice (preservesround curves). But, computing all US(β) may be hard (may requireexponential time !)

Reasonably-sounding result

An element of US(β) has the “simplest” normal form, so if β is reducible,elements of all US(β) preserves the simplest, a standard round curve.

This is true under some assumptions (Lee-Lee ’08), but is not true ingeneral: (think appropriate simple element, for example)

Drawback

The theorem says at least one element in US(β) is very nice (preservesround curves). But, computing all US(β) may be hard (may requireexponential time !)

Reasonably-sounding result

An element of US(β) has the “simplest” normal form, so if β is reducible,elements of all US(β) preserves the simplest, a standard round curve.

This is true under some assumptions (Lee-Lee ’08), but is not true ingeneral: (think appropriate simple element, for example)

Fast Nielsen-Thurston type via Garside theory

Theorem (Gonzalez-Meneses, Wiest ’11)

If β is reducible, then after taking m-th power βm for some m < n6, everyelement in α ∈ SC (βm) preserves either standard round curves or, almostround curves. (Here SC ⊂ US is a sliding circuit, a more refinement of theUltra summit set)

Round Almost Round

Conclusion

Having simple normal form (simple in algebraic prospect) = Having simplereduction curve (simple in geomteric prospect),

Fast Nielsen-Thurston type via Garside theory

Moreover, by applying linear bounded conjugator property

Theorem (Mazur-Minsky ’00, Tao ’13)

If x , y ∈ Bn are conjugate, then x = wxw−1, where the length of w ∈ Bn

is at most Constant C (n) · (length of x + y)))

We have (theoretically fast) algorithm:

Theorem (Calvez ’14)

By using Garside theory machinery, one can determine whether β isreducible or not in quadratic time.

Remark

Unfortunately, due to the lack of our knowledge of precise value of C (n),the algorithm in thw above theorem is not practical at this moment.

Questions

At this moment, our argument recognizes periodic and reducible braids.

Problem

Can we recoginze/understand pseudo-Anosov braid (dilatation, theirinvariant train-track) from Garside theory ?

A reasonably-sounding idea is that if α is pseudo-Anosov and β ∈ SS(α),then the invariant train-track of β is simple in some sense.

Remark

For a pseudo-Anosov braid β, then there exists m < n6 such that thenormal form of βm has certain nice property called rigidity.

I-5: Application to topology (2): Curvediagram and linear representation

Curve diagram

Using identification Bn∼= MCG (Dn), we can represent β ∈ Bn by the

(isotopy class of the) image of horizontal line Γ, called Curve Diagram.

(We often distinguish the first segment e of Γ connecting the boundaryand the first puncture, and define

Γβ = (Γ− e)β

Labelling of Curve diagram I: winding number labelling

Make curve diagram so that it has minimum vertical tangencies, andassign labelling (winding number labelling) as follows: if we turn clockwisedirection, add +1 and if we turn counter-clockwise direction, add −1

(i + 1)

Labelling of Curve diagram II: wall-crossing numberlabelling

Make curve diagram so that it has minimum intersection with walls(vertical line from punctures) and that near the puncture it is horizontal.Assign labelling wall crossing labelling by signed counting of intersectionswith walls (here we escape puncture in counter-clockwise direction).

i (i + 1)

(i + 1) i

Labelling of Curve diagram and Garside theory

Theorem (Wiest ’09)

1. max Winding number labelling on Γβ = sup(β)

2. min Winding number labelling on Γβ = inf(β)

(Classical Garside normal form measures “how many times the braid βwinds real axis”)

Theorem (I-Wiest ’12)

1. max Wall crossing number labelling on Γβ = sup ∗(β)

2. min Wall crossing number labelling on Γβ = inf ∗(β)

(Dual Garside normal form measures “how many times the image of thereal axis across the walls”)

Labelling of Curve diagram and Garside theory

Theorem (Wiest ’09)

1. max Winding number labelling on Γβ = sup(β)

2. min Winding number labelling on Γβ = inf(β)

(Classical Garside normal form measures “how many times the braid βwinds real axis”)

Theorem (I-Wiest ’12)

1. max Wall crossing number labelling on Γβ = sup ∗(β)

2. min Wall crossing number labelling on Γβ = inf ∗(β)

(Dual Garside normal form measures “how many times the image of thereal axis across the walls”)

Sketch of proofStrategy:

Multiply inverse of (dual) simple elements so that maximum labellingdecreases

This process provides an effective (fastest) way to make the braidtrivial by using (dual) simple elements⇒ it is the meaning of normal form!

Here we give a proof for dual case: we isotope curve diagram and wall sothat it has circular symmetry (wall-corssing labelling does not change).

Sketch of proof

The set of arcs in curve diagram with maximal wall-crossing labellingsuggests which dual simple element is needed to simplify the diagram: the“convex hull” of maximally labelled arcs provides the most economicaluntangling dual simple element.

Lawrence-Krammer-Bigelow representation

C : Configration space of two points in Dn

C = (z1, z2) ∈ D2n | z1 = z2/(z1, z2) ≡ z2, z1)

then H1(C ;Z) = Zn ⊕ Z =⊕⟨xi ⟩ ⊕ ⟨t⟩, where

xi : meridian of hypersurface z1 = i-th puncturet : meridian of hypersurface z1 = z2

Let π : C → C be the Z2-cover associated with the kernel of

α : π1(C )Hurewicz→ H1(C ;Z)→ Z2 ∼= ⟨x⟩ ⊕ ⟨t⟩ (xi 7→ x , t 7→ t).

H2(C ;Z) is a free Z[x±1, t±1]-module of rank(n2

C : Configration space of two points in Dn

C = (z1, z2) ∈ D2n | z1 = z2/(z1, z2) ≡ z2, z1)

then H1(C ;Z) = Zn ⊕ Z =⊕⟨xi ⟩ ⊕ ⟨t⟩, where

xi : meridian of hypersurface z1 = i-th puncturet : meridian of hypersurface z1 = z2

Let π : C → C be the Z2-cover associated with the kernel of

α : π1(C )Hurewicz→ H1(C ;Z)→ Z2 ∼= ⟨x⟩ ⊕ ⟨t⟩ (xi 7→ x , t 7→ t).

H2(C ;Z) is a free Z[x±1, t±1]-module of rank(n2

The braid group Bn = MCG (Dn) action on Dn induces an action on C (upto homotopy), so we get

ρLKB : Bn → GL(H2(C ;Z))

called the Lawrence-Krammer-Bigelow representation. By choosingappropriate basis vi ,j1≤i<j≤n coming from topology, the LKBrepresentation is given by

ρLKB(σi )(vj ,k) =

Fj ,k i ∈ j − 1, j , k − 1, kqFi ,k + (q2 − q)Fi ,j + (1− q)Fj ,k i = j − 1

Fj+1,k i = j = k − 1

qFj ,i + (1− q)Fj ,k + (q − q2)tFi ,k i = k − 1 = j

Fj ,k+1 i = k

−q2tFj ,k i = j = k − 1

Surprisingly, Lawrence-Krammer-Bigelow representation detects the normalforms.

Theorem (Krammer ’02, I-Wiest ’12)

For β ∈ Bn,

1. maxdegree of t in the matrix ρLKB(β) = sup(β).

2. mindegree of t in the matrix ρLKB(β) = inf(β)

3. maxdegree of q in the matrix ρLKB(β) = 2 sup ∗(β).

4. mindegree of q in the matrix ρLKB(β) = 2 inf ∗(β)

Corollary (Krammer, Bigelow ’02)

The Lawrence-Krammer-Bigelow representation is faithful – so, the braidgroups are linear.

Why LKB representation know the Garside structures ?

Compare the definition of α : π1(C )→ ⟨x⟩ ⊕ ⟨t⟩ with the definition oflabelling of curve diagram:

Labelling = Position of the lift of the curve ∼= variables q and t.

Quantum representation

By theory of quantum group, for a Uq(g)-module V , (quantum envelopingalgebra of semi-simple lie algebra g), we have a linear representation calledquantum representations

ρV : Bn → GL(V⊗n)

that is a q-deformation of permutation

ϕV : Sn → GL(V⊗n),

(i , i + 1)(v1 ⊗ · · · ⊗ vi−1 ⊗ vi ⊗ · · · ⊗ vn) = v1 ⊗ · · · ⊗ vi+1 ⊗ vi ⊗ vn

Quantum representations are important because they produces invariantsof knto and 3-manifolds, called Quantum invariants.

Quantum representation and invariants

Braids

ρVQuantum

representation

Closure // (Oriented) Links Surgery//

Quantuminvariant

Closed 3-manifolds

Quantuminvariant

GL(V⊗n)

“Trace′′ // C[q, q−1]q=e

2π√

Take linear sums// C

Quantum representation and Garside theory

Using KZ-equation argument (realizing quantum representation as certainmonodromy representation), one identifies “generic” quantum sl2representation with homological representation similar toLawrence-Krammer-Bigelow representation (Kohno,I, Jackson-Kerler).Then, we have:

Theorem (I. ’12)

For β ∈ Bn. the maximal and the minimal degree of weight variable in“Generic” quantum sl2-representation is equal to the constant multiples ofsup ∗(β) and inf ∗(β).

⇒ Quantum representation (quantum group) is also related to (dual)Garside structure.

Problems

Problem

Find a relationship between linear representations and the classical Garsidestructure: Conjecturally, it should be related to the quantum parameter q.

Problem

Find a relationship between quantum knots or3-manifold invariants (forexample, Jones polynomial) and Garside theory.

Problem

Find a direct, more conceptual understanding between quantumrepresentation and Garside structure.

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