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Garside structure and Dehornoy ordering of braid groups for topologist (mini-course I) Tetsuya Ito Combinatorial Link Homology Theories, Braids, and Contact Geometry Aug, 2014 Tetsuya Ito Braid calculus Sep , 2014 1 / 98 Contents

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

Tetsuya Ito Combinatorial Link Homology Theories, Braids, and Contact Geometry Aug, 2014

Tetsuya Ito Braid calculus Sep , 2014 1 / 98

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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 linear

representation

Tetsuya Ito Braid calculus Sep , 2014 2 / 98

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Introduction

Tetsuya Ito Braid calculus Sep , 2014 3 / 98

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Braid group

The n-strand braid group Bn = ⟨ σ1, . . . , σn−1 σiσjσi = σiσjσi, |i − j| = 1 σiσj = σjσi, |i − j| > 1 ⟩ .

C t = t = 1
  • i
1 2 i i + 1 n
  • 1
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 the kernel of π.

Tetsuya Ito Braid calculus Sep , 2014 4 / 98

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Braid group in topology (I) relation to knot theory

Alexander-Markov Theorem

{Braids}/conjugation,stabilization)

1:1

← → { Oriented links in S3}

  • 1

Tetsuya Ito Braid calculus Sep , 2014 5 / 98

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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}

  • 1

Tetsuya Ito Braid calculus Sep , 2014 6 / 98

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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 : Dn

Homeo

− → Dn | f |∂Dn = id}/{Isotopy}

t = t = 1 half Dehn-t wist

Tetsuya Ito Braid calculus Sep , 2014 7 / 98

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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 : Dn

Homeo

− → Dn | f |∂Dn = id}/{Isotopy}

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

t = t = 1 half Dehn-t wist

Tetsuya Ito Braid calculus Sep , 2014 7 / 98

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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 so { ΩUCn(C) = {Space of braids} π1(Cn(C)) = Pn, π1(UCn(C)) = Bn.

Tetsuya Ito Braid calculus Sep , 2014 8 / 98

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Braid group in topology (III) configuration space

A natural projection p : Cn(C) → Cn−1(C), p(z1, . . . , zn) → (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

Tetsuya Ito Braid calculus Sep , 2014 9 / 98

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Braid group in topology (III) configuration space

A natural projection p : Cn(C) → Cn−1(C), p(z1, . . . , zn) → (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)) · · · ).

Tetsuya Ito Braid calculus Sep , 2014 9 / 98

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Braid group in topology (IV) Hyperplane arrangement

Cn(C) is regarded as the complement of an hyperplane arrangement called the braid arrangement: For 1 ≤ i < j ≤ n, let Hi,j = Ker(zi − zj) ⊂ Cn, A = {Hi,j}1≤i<j≤n Then Cn(C) = M(A) = Cn − ∪

1≤i<j≤n

Hi,j,

Tetsuya Ito Braid calculus Sep , 2014 10 / 98

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Braid group in topology (IV) Hyperplane arrangement

Cn(C) is regarded as the complement of an hyperplane arrangement called the braid arrangement: For 1 ≤ i < j ≤ n, let Hi,j = Ker(zi − zj) ⊂ Cn, A = {Hi,j}1≤i<j≤n Then Cn(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.

Tetsuya Ito Braid calculus Sep , 2014 10 / 98

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Part I: Garside theory for braid groups

Tetsuya Ito Braid calculus Sep , 2014 11 / 98

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Word and conjugacy problem

Word/Conjugacy Problem

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

1 , . . . , 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.)

Tetsuya Ito Braid calculus Sep , 2014 12 / 98

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Word and conjugacy problem

Word/Conjugacy Problem

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

1 , . . . , 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.

Tetsuya Ito Braid calculus Sep , 2014 12 / 98

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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:

Tetsuya Ito Braid calculus Sep , 2014 13 / 98

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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:

Motto

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

▶ Dynamics (Nielsen-Thurston classification) ▶ Topology (infinite cyclic coverings) ▶ Algebra (quantum/homological representation) ▶ Dehornoy’s ordering

Tetsuya Ito Braid calculus Sep , 2014 13 / 98

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I-1: Toy model: Garside structure on Z2

Tetsuya Ito Braid calculus Sep , 2014 14 / 98

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Toy model: Garside structure on Z2

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

Tetsuya Ito Braid calculus Sep , 2014 15 / 98

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Toy model: Garside structure on Z2

G = Z2 = ⟨x, y⟩: Free abelian group of rank two P = {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, ∆}.

Tetsuya Ito Braid calculus Sep , 2014 15 / 98

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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

Tetsuya Ito Braid calculus Sep , 2014 16 / 98

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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−1

i−1 · · · s−1 1 ∆−p)β

(So normal form of β = xayb is actually written as: N(β) = { ∆ayb−a b ≥ a ∆bxa−b a ≥ b

Tetsuya Ito Braid calculus Sep , 2014 16 / 98

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What is the meaning of normal form ?

Idea

Normal form = path in the Cayley graph which approaches the destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ?

x y
  • Univ
ersit y Home

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

Tetsuya Ito Braid calculus Sep , 2014 17 / 98

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What is the meaning of normal form ?

Idea

Normal form = path in the Cayley graph which approaches destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ?

x y
  • Univ
ersit y Home y y xy y x

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

Tetsuya Ito Braid calculus Sep , 2014 18 / 98

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What is the meaning of normal form ?

Idea

Normal form = path in the Cayley graph which approaches the destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ?

x y
  • Univ
ersit y Home y y
  • v.s.
y y

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

Tetsuya Ito Braid calculus Sep , 2014 19 / 98

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What is the meaning of normal form ?

Idea

Normal form = path in the Cayley graph which approaches destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ?

x y
  • Univ
ersit y Home y y
  • v.s.
y y Normal form

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

Tetsuya Ito Braid calculus Sep , 2014 20 / 98

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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 sisi+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 sisi+1 with better one (tighten locally).

Tetsuya Ito Braid calculus Sep , 2014 21 / 98

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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 sisi+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 sisi+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)

Tetsuya Ito Braid calculus Sep , 2014 21 / 98

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Computing normal forms: example

x
  • y
xy y y x

Tetsuya Ito Braid calculus Sep , 2014 22 / 98

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Computing normal forms: example

x
  • y
xy y y x + xy y
  • Tetsuya Ito

Braid calculus Sep , 2014 23 / 98

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Computing normal forms: example

x
  • y
xy y
  • Tetsuya Ito

Braid calculus Sep , 2014 24 / 98

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Computing normal forms: example

x
  • y
xy y
  • +
xy y
  • Tetsuya Ito

Braid calculus Sep , 2014 25 / 98

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SLIDE 34

Computing normal forms: example

x
  • y
xy y
  • +
xy y y
  • Tetsuya Ito

Braid calculus Sep , 2014 26 / 98

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Computing normal forms: example

x
  • y
x y y
  • Tetsuya Ito

Braid calculus Sep , 2014 27 / 98

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Computing normal forms: example

x
  • y
xy y
  • +
xy y
  • Tetsuya Ito

Braid calculus Sep , 2014 28 / 98

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SLIDE 37

Computing normal forms: example

x
  • y
The b est hoi e
  • f
the rst path k The rst letter
  • f
the no rmal fo rm

Tetsuya Ito Braid calculus Sep , 2014 29 / 98

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Computing normal forms: example

x
  • y
xy y
  • +
xy y

Tetsuya Ito Braid calculus Sep , 2014 30 / 98

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SLIDE 39

Computing normal forms: example

x
  • y
xy y + xy y

Tetsuya Ito Braid calculus Sep , 2014 31 / 98

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Computing normal forms: example

x
  • y
x y y

Tetsuya Ito Braid calculus Sep , 2014 32 / 98

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Computing normal forms: example

x
  • y
x y y + x y y

Tetsuya Ito Braid calculus Sep , 2014 33 / 98

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Computing normal forms: example

x
  • y
The b est hoi e
  • f
the se ond path k The se ond letter
  • f
the no rmal fo rm

Tetsuya Ito Braid calculus Sep , 2014 34 / 98

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SLIDE 43

Computing normal forms: example

x
  • y
xy y

Tetsuya Ito Braid calculus Sep , 2014 35 / 98

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Computing normal forms: example

x
  • y
xy y

Tetsuya Ito Braid calculus Sep , 2014 36 / 98

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Computing normal forms: example

x
  • y
x y y + x y y

Tetsuya Ito Braid calculus Sep , 2014 37 / 98

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SLIDE 46

Computing normal forms: example

x
  • y
xy y

Tetsuya Ito Braid calculus Sep , 2014 38 / 98

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Computing normal forms: example

x
  • y
xy y +
  • y

Tetsuya Ito Braid calculus Sep , 2014 39 / 98

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SLIDE 48

Computing normal forms: example

x
  • y
The normal form is: y

Tetsuya Ito Braid calculus Sep , 2014 40 / 98

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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 performing

ℓ(ℓ−1) 2

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

Tetsuya Ito Braid calculus Sep , 2014 41 / 98

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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 performing

ℓ(ℓ−1) 2

= O(ℓ2) times of “local tightening” (replacing local roundabout route 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 look at the path of length two. This says

Conclusion’

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

Tetsuya Ito Braid calculus Sep , 2014 41 / 98

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SLIDE 51

I-2: Classical Garside structure

Tetsuya Ito Braid calculus Sep , 2014 42 / 98

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General idea of Garside structure

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

Tetsuya Ito Braid calculus Sep , 2014 43 / 98

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General idea of Garside structure

We want to generalize idea and method for “toy model” for more general and 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, . . . , }

  • f G.

▶ The notion of positive elements yields a subword ordering ≼:

α ≼ β

Def

⇐ ⇒ 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.

Tetsuya Ito Braid calculus Sep , 2014 43 / 98

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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

Tetsuya Ito Braid calculus Sep , 2014 44 / 98

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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 ≼ ∆.

Tetsuya Ito Braid calculus Sep , 2014 44 / 98

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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.

Tetsuya Ito Braid calculus Sep , 2014 45 / 98

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Simple braids

▶ ≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ3 2

  • Positive braids

▶ ∆ = σ1 ∨ σ2 ∨ · · · ∨ σn−1.

Tetsuya Ito Braid calculus Sep , 2014 46 / 98

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Simple braids

▶ ≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ3 2

  • 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)

Tetsuya Ito Braid calculus Sep , 2014 46 / 98

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SLIDE 59

Example: B3 case

∆ = (σ1σ2)σ1 = σ2σ1σ2, so [1, ∆] = {1, σ1, σ2, σ1σ2, σ2σ1, ∆} Simple braids: each strand positively crosses with other strands at most

  • nce.

Tetsuya Ito Braid calculus Sep , 2014 47 / 98

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Normal form

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

A braid β ∈ Bn admits the normal form N(β) = ∆px1x2 · · · xr (p ∈ Z, xi ∈ [1, ∆]) where

  • 1. ∆−pβ ∈ B+

n .

  • 2. xi = ∆ ∧ (x−1

i−1 · · · x−1 1 ∆−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.

Tetsuya Ito Braid calculus Sep , 2014 48 / 98

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How to compute normal form ?

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

Theorem (Elrifai-Morton, Thurston)

A word N′(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)

Tetsuya Ito Braid calculus Sep , 2014 49 / 98

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SLIDE 62

How to compute normal form ?

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

Strategy to get normal form

  • 1. Express β as a word of the form

β = ∆px1 · · · xr (p ∈ Z, xi ∈ [1, ∆])

Tetsuya Ito Braid calculus Sep , 2014 50 / 98

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SLIDE 63

How to compute normal form ?

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

Strategy to get normal form

  • 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 the center of Bn, so · · · σ−1

i

· · · = · · · ∆−2∆2σ−1

i

· · · = ∆−2 · · · (∆2σ−1

i

)

  • Positive braid

· · ·

Tetsuya Ito Braid calculus Sep , 2014 50 / 98

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SLIDE 64

How to compute normal form ?

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

Strategy to get normal form

  • 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 the center of Bn, so · · · σ−1

i

· · · = · · · ∆−2∆2σ−1

i

· · · = ∆−2 · · · (∆2σ−1

i

)

  • Positive braid

· · ·

  • 2. Apply local tightening repeatedly: for i = r, . . . , 1 rewrite each

sub-path xixi+1 so that it is a normal form xixi+1 = x′

i x′ i+1,

x′

i = (xixi+1) ∧ ∆

Tetsuya Ito Braid calculus Sep , 2014 50 / 98

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SLIDE 65

Simple example

Let us compute the normal form of a 3-braid β = (σ−1

2 )(σ1σ2)(σ2)(σ1σ2).

Tetsuya Ito Braid calculus Sep , 2014 51 / 98

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SLIDE 66

Simple example

Let us compute the normal form of a 3-braid β = (σ−1

2 )(σ1σ2)(σ2)(σ1σ2).

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

β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)

Tetsuya Ito Braid calculus Sep , 2014 51 / 98

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SLIDE 67

Simple example

Let us compute the normal form of a 3-braid β = (σ−1

2 )(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

Tetsuya Ito Braid calculus Sep , 2014 51 / 98

slide-68
SLIDE 68

Simple example: local tightening

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

slide-69
SLIDE 69

Simple example: local tightening

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2) (σ2)(σ1σ2) ∧ ∆ = ∆, so β′ = (σ1σ2)(σ1σ2)(∆).

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

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SLIDE 70

Simple example: local tightening

β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2) (σ2)(σ1σ2) ∧ ∆ = ∆, so β′ = (σ1σ2)(σ1σ2)(∆). (σ1σ2)(∆) ∧ ∆ = ∆, and (σ1σ2)(∆) = (∆)(σ2σ1), so β′ = (σ1σ2)(∆)(σ2σ1)

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

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SLIDE 71

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)

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

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SLIDE 72

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.

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

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SLIDE 73

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)

Tetsuya Ito Braid calculus Sep , 2014 52 / 98

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SLIDE 74

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)

Tetsuya Ito Braid calculus Sep , 2014 53 / 98

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SLIDE 75

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

F ( 2
  • 1
  • 3
) = f 1 ;
  • 3
g S S ( 1
  • 2
  • 3
) = f 1 g F
  • r es
to have se ond rossings b et w een t w
  • strands

Tetsuya Ito Braid calculus Sep , 2014 54 / 98

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SLIDE 76

Geodesic property

Lemma

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

Tetsuya Ito Braid calculus Sep , 2014 55 / 98

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SLIDE 77

Geodesic property

Lemma

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 simple braids [1, ∆] is ℓ[1,∆](β) = max{sup(β), 0} − min{inf(β), 0}.

Tetsuya Ito Braid calculus Sep , 2014 55 / 98

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SLIDE 78

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. In particular, {Set of normal forms}

1:1

← → {Path of certain graph (automata)}

Tetsuya Ito Braid calculus Sep , 2014 56 / 98

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SLIDE 79

Example: Automata for B3

  • 1
  • 1
  • 2
  • 1
  • 1
  • 1
  • 2
  • 2
  • 1
  • 2
  • 1
  • 2
  • 2
  • 1

Tetsuya Ito Braid calculus Sep , 2014 57 / 98

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SLIDE 80

Example: Automata for B3

  • 1
  • 1
  • 2
  • 1
  • 1
  • 1
  • 2
  • 2
  • 1
  • 2
  • 1
  • 2
  • 2
  • 1

Normal form N(β) = ∆−1∆−1(σ2σ1)(σ1σ2)(σ2σ1)

Tetsuya Ito Braid calculus Sep , 2014 58 / 98

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SLIDE 81

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 forms among its conjugacy class, called ... summit set. S(α) = { β β is conjugate to α, with the “simplest”N(β) +“Additional requirements” } Then, S(α) = S(α′) ⇐ ⇒ α and β are conjugate

Tetsuya Ito Braid calculus Sep , 2014 59 / 98

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SLIDE 82

Conjugacy problem (II)

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

N ( ) =
  • p
x 1
  • x
r 1 x r =
  • p
x 1
  • p
  • p
x 2
  • x
r x r
  • p
x 1
  • x
r 1
  • p
x 2
  • x
r 1
  • p
x 1
  • p
k
  • p
( p x r
  • p
)x 1
  • x
r 1
  • p
x 1
  • x
r : simpler normal form de y ling y ling

It may happen p′ > p or r′ < r

Tetsuya Ito Braid calculus Sep , 2014 60 / 98

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SLIDE 83

Conjugacy problem (II)

Theorem (Garside, ElRifai-Morton, Gebhardt, Gonz´ alez-Meneses)

Let α ∈ Bn.

  • 1. By applying cycling and decylings finitely many times, we can find
  • ne element in S(α).
  • 2. Staring from one element β ∈ S(α), by repeatedly computing the

conjugate of β by simple elements, we can find all elements of S(α): In particular, we have an algorithm to solve the conjugacy decision and problem (determine α ∼conj α′) and the conjugacy search problem (find β such that α = β−1α′β).

Tetsuya Ito Braid calculus Sep , 2014 61 / 98

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SLIDE 84

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

The super summit set SS(α) = { β β is conjugate to α with maximal inf, minimum sup } The ultra summit set US(α) = {β ∈ SS(α) | closed under cycling operation }

US SS

Tetsuya Ito Braid calculus Sep , 2014 62 / 98

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SLIDE 85

Conjugacy problem (III)

Using idea of summit set, we can solve the conjugacy problem (but in time O(elength), in general):

▶ Computing a normal form is easy (done in polynomial time). ▶ Starting from α, finding one element of S(α) is (conjecturally) done

in polynomial time.

▶ Size of S(α) might be quite huge – the size of S(α) might be

O(elength) (So computing whole S(α) might require exponential times...)

Tetsuya Ito Braid calculus Sep , 2014 63 / 98

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SLIDE 86

Conjugacy problem (III)

Using idea of summit set, we can solve the conjugacy problem (but in time O(elength), in general):

▶ Computing a normal form is easy (done in polynomial time). ▶ Starting from α, finding one element of S(α) is (conjecturally) done

in polynomial time.

▶ Size of S(α) might be quite huge – the size of S(α) might be

O(elength) (So computing whole S(α) might require exponential times...)

Problem

Find polynomial time algorithm for conjugacy problem of braids.

Problem

Understand the structure of summit sets.

Tetsuya Ito Braid calculus Sep , 2014 63 / 98

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SLIDE 87

I-3: Dual Garside structure

Tetsuya Ito Braid calculus Sep , 2014 64 / 98

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SLIDE 88

Dual Garside structure

The braid group has another Garside structure called dual Garside structure, 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,j}1≤i<j≤n is called a dual Garside generator (Birman-Ko-Lee generator or band generator).

1 i j i j n a i;j t wisted band

Tetsuya Ito Braid calculus Sep , 2014 65 / 98

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SLIDE 89

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

Tetsuya Ito Braid calculus Sep , 2014 66 / 98

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SLIDE 90

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

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} ▶ ai,j ≼∗ δ for all 1 ≤ i < j ≤ n.

Tetsuya Ito Braid calculus Sep , 2014 66 / 98

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SLIDE 91

Dual Garside structure

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 normal form) N∗(β) = δpd1d2 · · · dr (p ∈ Z, xi ∈ [1, δ]) which is characterized by

  • 1. p = min{n ∈ Z | δnβ ∈ B+∗

n }

  • 2. xi = δ ∧∗ (d−1

i−1 · · · d−1 1 δ−pβ).

We define the dual supremum, dual infimum of β by sup ∗(β) = p + r, inf ∗(β) = p

Tetsuya Ito Braid calculus Sep , 2014 67 / 98

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SLIDE 92

Dual Garside structure

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 word with respect to the length ℓ[1,δ]. In particular, ℓ[1,δ](β) = max{sup ∗(β), 0} − min{inf ∗(β), 0}. By the similar method, one can use dual normal form to solve the conjugacy problem.

Tetsuya Ito Braid calculus Sep , 2014 68 / 98

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SLIDE 93

Dual Garside structure

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

Tetsuya Ito Braid calculus Sep , 2014 69 / 98

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SLIDE 94

Dual Garside structure

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:

Tetsuya Ito Braid calculus Sep , 2014 69 / 98

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SLIDE 95

A geometric understanding of dual simple elements

Let us identify Bn with MCG(Dn). Then,

Proposition (Bessis)

{Set of convex polygons in Dn}

1:1

← → [1, δ] (Convex polygons is understood as non-crossing partition of n-points)

1 2 3 4 5 6 7 8 (a 1;3 )(a 4;5 a 5;7 a 7;8 )

Tetsuya Ito Braid calculus Sep , 2014 70 / 98

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SLIDE 96

A geometric understanding of the normal form condition

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

Proposition

For x, y ∈ [1, δ], δ ∧∗ (xy) = x ⇐ ⇒ Corresponding convex polygons x are “linked” to y y x

Linked Not Linked

x x y y

Tetsuya Ito Braid calculus Sep , 2014 71 / 98

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SLIDE 97

Open problem

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 numerical correspondence of several data of the Garside structures (numbers of atoms, simple elements, ...) and there is no theoretical “duality” at all !

Tetsuya Ito Braid calculus Sep , 2014 72 / 98

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SLIDE 98

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

Tetsuya Ito Braid calculus Sep , 2014 73 / 98

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SLIDE 99

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 three types: Periodic, reducible, pseudo-Anosov

Tetsuya Ito Braid calculus Sep , 2014 74 / 98

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SLIDE 100

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 three types: Periodic, reducible, pseudo-Anosov 1: Periodic βn = ∆2m for some n, m ∈ Z (i.e., Powers of β = Dehn twists along ∂Dn)

Tetsuya Ito Braid calculus Sep , 2014 74 / 98

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SLIDE 101

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 three types: 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 and is not isotopic to ∂Dn)

Tetsuya Ito Braid calculus Sep , 2014 74 / 98

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SLIDE 102

Nielsen-Thurston theory

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

Tetsuya Ito Braid calculus Sep , 2014 75 / 98

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SLIDE 103

Nielsen-Thurston theory

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 ?

Tetsuya Ito Braid calculus Sep , 2014 76 / 98

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SLIDE 104

Nielsen-Thurston theory

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 a solution of this problem (Bestvina-Handel algorithm). Now, Garside theory (normal form) provides alternative solution !

Tetsuya Ito Braid calculus Sep , 2014 76 / 98

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SLIDE 105

Nielsen-Thurston type via Garside theory

Recognizing a periodic braid is easy:

Theorem (Eilenberg, Ker´ ekj´ art´

  • )

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 preserve very,very,very complicated “simple” (so it is not simple – rather complex !!!) closed curve.

Tetsuya Ito Braid calculus Sep , 2014 77 / 98

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SLIDE 106

Nielsen-Thurston type via Garside theory

Recognizing a periodic braid is easy:

Theorem (Eilenberg, Ker´ ekj´ art´

  • )

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 preserve very,very,very complicated “simple” (so it is not simple – rather complex !!!) closed curve.

Idea

Assume β is reducible. If N(β) is simple among its conjugacy class, then β preserves “simple” (not complicated, near “standard”) simple closed curves. Simple normal form ⇐ ⇒ Preserving “simple” simple closed curve

Tetsuya Ito Braid calculus Sep , 2014 77 / 98

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SLIDE 107

Easy, but informative observation

Observation

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

Tetsuya Ito Braid calculus Sep , 2014 78 / 98

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SLIDE 108

Nielsen-Thurston type via Garside theory

Theorem (Barnadete-Nitecki-Guti´ errez ’95)

If β is reducible, then there exists α ∈ US(β) ⊂ SS(β) such that α preserves standard a round curve. Thus by computing US(β) or SS(β), we can 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.

Tetsuya Ito Braid calculus Sep , 2014 79 / 98

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SLIDE 109

Nielsen-Thurston type via Garside theory

Drawback

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

Tetsuya Ito Braid calculus Sep , 2014 80 / 98

slide-110
SLIDE 110

Nielsen-Thurston type via Garside theory

Drawback

The theorem says at least one element in US(β) is very nice (preserves round curves). But, computing all US(β) may be hard (may require exponential 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 in general: (think appropriate simple element, for example)

Tetsuya Ito Braid calculus Sep , 2014 80 / 98

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SLIDE 111

Fast Nielsen-Thurston type via Garside theory

Theorem (Gonz´ alez-Meneses, Wiest ’11)

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

Conclusion

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

Tetsuya Ito Braid calculus Sep , 2014 81 / 98

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SLIDE 112

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 is reducible 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.

Tetsuya Ito Braid calculus Sep , 2014 82 / 98

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SLIDE 113

Questions

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

Problem

Can we recoginze/understand pseudo-Anosov braid (dilatation, their invariant 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 the normal form of βm has certain nice property called rigidity.

Tetsuya Ito Braid calculus Sep , 2014 83 / 98

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SLIDE 114

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

Tetsuya Ito Braid calculus Sep , 2014 84 / 98

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SLIDE 115

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.

  • 1
  • 2
() 1
  • 2

(We often distinguish the first segment e of Γ connecting the boundary and the first puncture, and define Γβ = (Γ − e)β

Tetsuya Ito Braid calculus Sep , 2014 85 / 98

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SLIDE 116

Labelling of Curve diagram I: winding number labelling

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

i (i + 1) i (i + 1) (i + 1) i i (i + 1) 1 1

Tetsuya Ito Braid calculus Sep , 2014 86 / 98

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SLIDE 117

Labelling of Curve diagram II: wall-crossing number labelling

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 intersections with walls (here we escape puncture in counter-clockwise direction).

i (i + 1) (i + 1) i 1 2 2 1 1

Tetsuya Ito Braid calculus Sep , 2014 87 / 98

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SLIDE 118

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 β winds real axis”)

Tetsuya Ito Braid calculus Sep , 2014 88 / 98

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SLIDE 119

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 β winds 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 the real axis across the walls”)

Tetsuya Ito Braid calculus Sep , 2014 88 / 98

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SLIDE 120

Sketch of proof

Strategy:

▶ Multiply inverse of (dual) simple elements so that maximum labelling

decreases

Tetsuya Ito Braid calculus Sep , 2014 89 / 98

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SLIDE 121

Sketch of proof

Strategy:

▶ Multiply inverse of (dual) simple elements so that maximum labelling

decreases

▶ This process provides an effective (fastest) way to make the braid

trivial by using (dual) simple elements ⇒ it is the meaning of normal form!

Tetsuya Ito Braid calculus Sep , 2014 89 / 98

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SLIDE 122

Sketch of proof

Strategy:

▶ Multiply inverse of (dual) simple elements so that maximum labelling

decreases

▶ This process provides an effective (fastest) way to make the braid

trivial 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 so that it has circular symmetry (wall-corssing labelling does not change).

Tetsuya Ito Braid calculus Sep , 2014 89 / 98

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SLIDE 123

Sketch of proof

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

Tetsuya Ito Braid calculus Sep , 2014 90 / 98

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SLIDE 124

Lawrence-Krammer-Bigelow representation

C : Configration space of two points in Dn C = {(z1, z2) ∈ D2

n | z1 ̸= z2}/(z1, z2) ≡ z2, z1)

then H1(C; Z) = Zn ⊕ Z = ⊕⟨xi⟩ ⊕ ⟨t⟩, where { xi : meridian of hypersurface {z1 = i-th puncture} t : meridian of hypersurface {z1 = z2}

Tetsuya Ito Braid calculus Sep , 2014 91 / 98

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SLIDE 125

Lawrence-Krammer-Bigelow representation

C : Configration space of two points in Dn C = {(z1, z2) ∈ D2

n | z1 ̸= z2}/(z1, z2) ≡ z2, z1)

then H1(C; Z) = Zn ⊕ Z = ⊕⟨xi⟩ ⊕ ⟨t⟩, where { xi : meridian of hypersurface {z1 = i-th puncture} t : 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 → x, t → t). H2( C; Z) is a free Z[x±1, t±1]-module of rank (n

2

) .

Tetsuya Ito Braid calculus Sep , 2014 91 / 98

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SLIDE 126

Lawrence-Krammer-Bigelow representation

The braid group Bn = MCG(Dn) action on Dn induces an action on C (up to homotopy), so we get ρLKB : Bn → GL(H2( C; Z)) called the Lawrence-Krammer-Bigelow representation. By choosing appropriate basis {vi,j}1≤i<j≤n coming from topology, the LKB representation is given by ρLKB(σi)(vj,k) =                      Fj,k i ̸∈ {j − 1, j, k − 1, k} qFi,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

Tetsuya Ito Braid calculus Sep , 2014 92 / 98

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Lawrence-Krammer-Bigelow representation

Surprisingly, Lawrence-Krammer-Bigelow representation detects the normal forms.

Theorem (Krammer ’02, I-Wiest ’12)

For β ∈ Bn,

  • 1. max{degree of t in the matrix ρLKB(β)} = sup(β).
  • 2. min{degree of t in the matrix ρLKB(β)} = inf(β)
  • 3. max{degree of q in the matrix ρLKB(β)} = 2 sup ∗(β).
  • 4. min{degree of q in the matrix ρLKB(β)} = 2 inf ∗(β)

Corollary (Krammer, Bigelow ’02)

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

Tetsuya Ito Braid calculus Sep , 2014 93 / 98

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Why LKB representation know the Garside structures ?

Compare the definition of α : π1( C) → ⟨x⟩ ⊕ ⟨t⟩ with the definition of labelling of curve diagram: Labelling = Position of the lift of the curve ∼ = variables q and t.

D n 1 1
  • e
  • f
D n q q

Tetsuya Ito Braid calculus Sep , 2014 94 / 98

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SLIDE 129

Quantum representation

By theory of quantum group, for a Uq(g)-module V , (quantum enveloping algebra of semi-simple lie algebra g), we have a linear representation called quantum 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 invariants

  • f knto and 3-manifolds, called Quantum invariants.

Tetsuya Ito Braid calculus Sep , 2014 95 / 98

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Quantum representation and invariants

{Braids}

ρV

Quantum representation

  • Closure {(Oriented) Links }

Surgery

  • Quantum

invariant

  • {Closed 3-manifolds}

Quantum invariant

  • GL(V ⊗n)

“Trace′′

C[q, q−1]

q=e

2π√−1 N

Take linear sums

C

Tetsuya Ito Braid calculus Sep , 2014 96 / 98

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Quantum representation and Garside theory

Using KZ-equation argument (realizing quantum representation as certain monodromy representation), one identifies “generic” quantum sl2 representation with homological representation similar to Lawrence-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 of sup ∗(β) and inf ∗(β). ⇒ Quantum representation (quantum group) is also related to (dual) Garside structure.

Tetsuya Ito Braid calculus Sep , 2014 97 / 98

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Problems

Problem

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

Problem

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

Problem

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

Tetsuya Ito Braid calculus Sep , 2014 98 / 98