Garside Structure and Dehornoy Ordering of Braid Groups for Topologist (Mini-Course II)

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

Tetsuya Ito

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

Tetsuya Ito Braid calculus Aug , 2014 1 / 64 ▶ Part II: The Dehornoy ordering ▶ II-1: Dehornoy’s ordering: deﬁnition ▶ II-2: How to compute Dehornoy’s ordering ? ▶ II-3: Application (1): Knot theory ▶ II-4: Application (2): FDTC and contact geometry ▶ II-5: The Dehornoy ordering and Garside theory

Tetsuya Ito Braid calculus Aug , 2014 2 / 64 Part II Dehornoy’s ordering

Tetsuya Ito Braid calculus Aug , 2014 3 / 64 II-1: Dehornoy’s ordering: deﬁnition

Tetsuya Ito Braid calculus Aug , 2014 4 / 64 σ-positive word

Deﬁnition { ±1 ±1 } W : word over σ1 , . . . , σn−1   contains at least one σi σi − positive ⇐⇒ W ±1 ±1 −1 does not contain σ1 , . . . , σi−1, σi . W is −1  contains at least one σi σi − negative ⇐⇒ W ±1 ±1 does not contain σ1 , . . . , σi−1, σi .

A braid β ∈ Bn is { σ-positive ⇐⇒ β admits σi -positive word representatives for some i.

σ-negative ⇐⇒ β admits σi -positive word representatives for some i.

(Note: β is σ-positive ⇐⇒ β−1 is σ-negative.)

Tetsuya Ito Braid calculus Aug , 2014 5 / 64 ▶ −1 W = σ1σ2σ1 : Neither σ-positive nor σ-negative word. ▶ −1 As a braid, β = σ1σ2σ1 is σ-positive:

−1 −1 σ1σ2σ1 = σ2 σ1σ2 : σ1-positive word

At ﬁrst glance, a σ-positive/negative braid seems to be very special and it looks hard to know whether a given braid is σ-positive or not (even for 3-braids).

Quiz: 3 −2 −2 2 −7 Is a braid σ1σ2σ1 σ2 σ1σ2σ1σ2 σ-positive ?

σ-positive word

Examples:

▶ −1 W = σ2σ3 : σ2-positive word

Tetsuya Ito Braid calculus Aug , 2014 6 / 64 ▶ −1 As a braid, β = σ1σ2σ1 is σ-positive:

−1 −1 σ1σ2σ1 = σ2 σ1σ2 : σ1-positive word

At ﬁrst glance, a σ-positive/negative braid seems to be very special and it looks hard to know whether a given braid is σ-positive or not (even for 3-braids).

Quiz: 3 −2 −2 2 −7 Is a braid σ1σ2σ1 σ2 σ1σ2σ1σ2 σ-positive ?

σ-positive word

Examples:

▶ −1 W = σ2σ3 : σ2-positive word ▶ −1 W = σ1σ2σ1 : Neither σ-positive nor σ-negative word.

Tetsuya Ito Braid calculus Aug , 2014 6 / 64 σ-positive word

Examples:

▶ −1 W = σ2σ3 : σ2-positive word ▶ −1 W = σ1σ2σ1 : Neither σ-positive nor σ-negative word. ▶ −1 As a braid, β = σ1σ2σ1 is σ-positive:

−1 −1 σ1σ2σ1 = σ2 σ1σ2 : σ1-positive word

At ﬁrst glance, a σ-positive/negative braid seems to be very special and it looks hard to know whether a given braid is σ-positive or not (even for 3-braids).

Quiz: 3 −2 −2 2 −7 Is a braid σ1σ2σ1 σ2 σ1σ2σ1σ2 σ-positive ?

Tetsuya Ito Braid calculus Aug , 2014 6 / 64 Dehornoy’s ordering: Algebraic deﬁnition

Theorem-Deﬁnition (Denornoy ’94)

Deﬁne the relation

−1 α

Then

Corollary 1. If β ≠ 1, either β or β−1 is σ-positive. =⇒ σ-positive/negative words are ubiquitous !!! 2. A σ-positive word represents a non-trivial braid.

Tetsuya Ito Braid calculus Aug , 2014 7 / 64 1. Bn is torsion-free:

For non trivial β, either 1 D β. Then { 2 i 1 D β ⇒ 1 >D β >D β >D ··· >D β >D ···

2. The group ring ZBn has no zero-divisors. (i.e. for x, y ∈ ZBn, xy ≠ 0 if x, y ≠ 0)

Consequence of orderability

From the simple fact that Bn admits a left-ordering

Tetsuya Ito Braid calculus Aug , 2014 8 / 64 2. The group ring ZBn has no zero-divisors. (i.e. for x, y ∈ ZBn, xy ≠ 0 if x, y ≠ 0)

Consequence of orderability

From the simple fact that Bn admits a left-ordering

1. Bn is torsion-free:

For non trivial β, either 1 D β. Then { 2 i 1 D β ⇒ 1 >D β >D β >D ··· >D β >D ···

Tetsuya Ito Braid calculus Aug , 2014 8 / 64 Consequence of orderability

From the simple fact that Bn admits a left-ordering

1. Bn is torsion-free:

For non trivial β, either 1 D β. Then { 2 i 1 D β ⇒ 1 >D β >D β >D ··· >D β >D ···

2. The group ring ZBn has no zero-divisors. (i.e. for x, y ∈ ZBn, xy ≠ 0 if x, y ≠ 0)

Tetsuya Ito Braid calculus Aug , 2014 8 / 64 k 2. σ2 0. −1 3. 1

−1 −1 −1 −1 1 = (σ1σ2σ1)1(σ1σ2σ1) >D (σ1σ2σ1)(σ1σ2 )(σ1σ2σ1) = σ2σ1

so, the relation

4. If β is a positive braid (product of σ1, . . . , σn−1), then 1

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

1. 1

Tetsuya Ito Braid calculus Aug , 2014 9 / 64 −1 3. 1

−1 −1 −1 −1 1 = (σ1σ2σ1)1(σ1σ2σ1) >D (σ1σ2σ1)(σ1σ2 )(σ1σ2σ1) = σ2σ1

so, the relation

4. If β is a positive braid (product of σ1, . . . , σn−1), then 1

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

1. 1 0.

Tetsuya Ito Braid calculus Aug , 2014 9 / 64 , but

−1 −1 −1 −1 1 = (σ1σ2σ1)1(σ1σ2σ1) >D (σ1σ2σ1)(σ1σ2 )(σ1σ2σ1) = σ2σ1

so, the relation

4. If β is a positive braid (product of σ1, . . . , σn−1), then 1

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

1. 1 0. −1 3. 1

Tetsuya Ito Braid calculus Aug , 2014 9 / 64 4. If β is a positive braid (product of σ1, . . . , σn−1), then 1

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

1. 1 0. −1 3. 1

−1 −1 −1 −1 1 = (σ1σ2σ1)1(σ1σ2σ1) >D (σ1σ2σ1)(σ1σ2 )(σ1σ2σ1) = σ2σ1

so, the relation

Tetsuya Ito Braid calculus Aug , 2014 9 / 64 Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

1. 1 0. −1 3. 1

−1 −1 −1 −1 1 = (σ1σ2σ1)1(σ1σ2σ1) >D (σ1σ2σ1)(σ1σ2 )(σ1σ2σ1) = σ2σ1

so, the relation

4. If β is a positive braid (product of σ1, . . . , σn−1), then 1

Tetsuya Ito Braid calculus Aug , 2014 9 / 64 Property S (Subword Property)

Theorem (Property S, Lavor ’96) −1 For any β ∈ Bn, 1

Property S has several consequences: Corollary

1. αβ

2. A band generator is

∗ α ≼ β ⇒ α

Tetsuya Ito Braid calculus Aug , 2014 10 / 64 An existence of

Can we compute the

Well-orderedness An important consequence of the Property S is: Theorem (Lavar, Burckel, Dehornoy, Fromentin, I) + 1. The restriction of

Tetsuya Ito Braid calculus Aug , 2014 11 / 64 Well-orderedness An important consequence of the Property S is: Theorem (Lavar, Burckel, Dehornoy, Fromentin, I) + 1. The restriction of

An existence of

Can we compute the

Tetsuya Ito Braid calculus Aug , 2014 11 / 64 Theorem (Fromentin ’11) For every braid β admits a σ-positive/negative word expression which is a quasi-geodesic: { ±1 ±1 } For a given n-braid β with length ℓ (with respect to σ1 , . . . , σn−1 ), we can ﬁnd a σ-positive/negative word expression of β with length at most 6(n − 1)2ℓ.

(Remark: In the proof of this theorem, dual Garside structure is eﬀectively used ! )

Short σ-positive word representatives ?

The Dehornoy ordering says that every non-trivial braid is represented by a σ-positive or a σ-negative word. Question Is there a “good” σ-positive/-negative word representatives ?

Tetsuya Ito Braid calculus Aug , 2014 12 / 64 Short σ-positive word representatives ?

The Dehornoy ordering says that every non-trivial braid is represented by a σ-positive or a σ-negative word. Question Is there a “good” σ-positive/-negative word representatives ?

Theorem (Fromentin ’11) For every braid β admits a σ-positive/negative word expression which is a quasi-geodesic: { ±1 ±1 } For a given n-braid β with length ℓ (with respect to σ1 , . . . , σn−1 ), we can ﬁnd a σ-positive/negative word expression of β with length at most 6(n − 1)2ℓ.

(Remark: In the proof of this theorem, dual Garside structure is eﬀectively used ! )

Tetsuya Ito Braid calculus Aug , 2014 12 / 64 Dehornoy’s ordering: geometric view (1)

Theorem (Fenn-Greene-Rourke-Rolfsen-Wiest ’99) β(Γ)“moves the left side” of α(Γ) α < β ⇐⇒

D when we put β(Γ) and α(Γ) intersect minimally

´ µ

ÐefØ ×ide

ÖighØ ×ide

Tetsuya Ito Braid calculus Aug , 2014 13 / 64 Dehornoy’s ordering: geometric view (2)

[Skecth of proof:] (⇒) Assume β is represented by a σ1-positive word: ··· { ±1 ±1 } β = Wnσ1 W1σ1W0 (Wi : word over σ2 , . . . , σn−1 )

Let us look at the image of (the ﬁrst segment of) Γ :

Ï Ï Ï

¾ ½ ¼

½ ½ ½

β(Γ) remains to move left directions of Γ.

Tetsuya Ito Braid calculus Aug , 2014 14 / 64 Dehornoy’s ordering: geometric view (2)

(⇐) If β(Γ) moves the left direcition of Γ, we can simplify β(Γ) by

applying braid containing no σ1:

½ ½

¾ ½

Tetsuya Ito Braid calculus Aug , 2014 15 / 64 ▶ Algorithm to determine 1

Important prospect A reasonable procedure of simplifying curve diagrams yields a connection of topology/geometry of braids and algebraic structure (Garside normal form, Dehornoy ordering) of braid groups. A curve diagram is a deep and important object than our ﬁrst impression (although it is very simple) !!! (There might be other nice property of braids read from curve diagrams...)

Dehornoy’s ordering: geometric view (3) Application Geometric point of view yields: ▶ A simple proof of Property S.

Tetsuya Ito Braid calculus Aug , 2014 16 / 64 ▶ Algorithm to ﬁnd σ-positive representative word of β if 1

Dehornoy’s ordering: geometric view (3) Application Geometric point of view yields: ▶ A simple proof of Property S. ▶ Algorithm to determine 1

Tetsuya Ito Braid calculus Aug , 2014 16 / 64 Important prospect A reasonable procedure of simplifying curve diagrams yields a connection of topology/geometry of braids and algebraic structure (Garside normal form, Dehornoy ordering) of braid groups. A curve diagram is a deep and important object than our ﬁrst impression (although it is very simple) !!! (There might be other nice property of braids read from curve diagrams...)

Dehornoy’s ordering: geometric view (3) Application Geometric point of view yields: ▶ A simple proof of Property S. ▶ Algorithm to determine 1

Tetsuya Ito Braid calculus Aug , 2014 16 / 64 Dehornoy’s ordering: geometric view (3) Application Geometric point of view yields: ▶ A simple proof of Property S. ▶ Algorithm to determine 1

Tetsuya Ito Braid calculus Aug , 2014 16 / 64 Solution The hyperbolic geometry is useful to give the “ best” representative of curves: Equip hyperbolic structure of Dn. Then, ▶ By isotopy, one can realize every non-trivial curve as geodesic. ▶ Two geodesics on hyperbolic surface minimally intersect. ▶ 2 In the universal covering of Dn ⊂ H , geodesic is easy to see: if we put the base point in the center of disc model of H2, geodesic is just a straight line.

Dehornoy’s ordering: More schematic geometric view

We want to remove “up to isotopy” in curve diagram deﬁnition of

How to put two curves on Dn so that they intersect minimally (i.e. ﬁnd the “best” isotopy class)?

Tetsuya Ito Braid calculus Aug , 2014 17 / 64 ▶ By isotopy, one can realize every non-trivial curve as geodesic. ▶ Two geodesics on hyperbolic surface minimally intersect. ▶ 2 In the universal covering of Dn ⊂ H , geodesic is easy to see: if we put the base point in the center of disc model of H2, geodesic is just a straight line.

Dehornoy’s ordering: More schematic geometric view

We want to remove “up to isotopy” in curve diagram deﬁnition of

Tetsuya Ito Braid calculus Aug , 2014 17 / 64 Dehornoy’s ordering: More schematic geometric view

We want to remove “up to isotopy” in curve diagram deﬁnition of

How to put two curves on Dn so that they intersect minimally (i.e. ﬁnd the “best” isotopy class)? Solution The hyperbolic geometry is useful to give the “ best” representative of curves: Equip hyperbolic structure of Dn. Then, ▶ By isotopy, one can realize every non-trivial curve as geodesic. ▶ Two geodesics on hyperbolic surface minimally intersect. ▶ 2 In the universal covering of Dn ⊂ H , geodesic is easy to see: if we put the base point in the center of disc model of H2, geodesic is just a straight line.

Tetsuya Ito Braid calculus Aug , 2014 17 / 64 Dehornoy’s ordering: More schematic geometric view

(Figure borrowed from Short-Wiest’s paper) Tetsuya Ito Braid calculus Aug , 2014 18 / 64 Nielsen-Thurston type ordering

Using hyperbolic geometry construction, we can generalize the Dehornoy ordering for mapping class group of surface with non-empty boundary:

S: Hyperbolic Surface with non-empty geodesic boundary H2 ⊃ Se →π S: Universal covering

By considering the lifted action on boundary at inﬁnity (which does not depend on a choice of representative homeomorphism) we get an injective homomorphism Θ: MCG(S) → Homeo+(R) called the Nielsen-Thurston map. Remark The map Θ is not canonical – it may depends on various intermediate choices (hyperbolic metric etc...)

Tetsuya Ito Braid calculus Aug , 2014 19 / 64 Nielsen-Thurston type ordering

Deﬁnition

Take an ordered, countable dense subset {x1, x2,...} of R. For ϕ, ψ ∈ MCG(S), deﬁne { [Θ(ϕ)] (x ) = [Θ(ψ)] (x ) i = 1,..., j − 1 ϕ < ψ ⇐⇒ ∃j s.t. i i [Θ(ϕ)] (xj ) < [Θ(ψ)] (xj )

This deﬁnes a left ordering of MCG(S), called the Nielsen-Thurston type orderings.

Remark The Dehornoy ordering is regarded as a special one of the Nielsen-Thurston type ordering.

Tetsuya Ito Braid calculus Aug , 2014 20 / 64 II -2 Technique to compute Dehornoy ordering (handle reduction)

Tetsuya Ito Braid calculus Aug , 2014 21 / 64 Idea: −1 By modifying the word of the form σ1(σ1-free word)σ1 (this is a bad sequence) we may get σ1-positive/negative word.

Handle reduction

How to determine 1

−1 −1 σ1σ2σ1 = σ2 σ1σ2 (σ1-positive word)

k −1 −1 k More generally, σ1σ2 σ1 = σ2 σ1 σ2

Tetsuya Ito Braid calculus Aug , 2014 22 / 64 Handle reduction

How to determine 1

−1 −1 σ1σ2σ1 = σ2 σ1σ2 (σ1-positive word)

k −1 −1 k More generally, σ1σ2 σ1 = σ2 σ1 σ2

Idea: −1 By modifying the word of the form σ1(σ1-free word)σ1 (this is a bad sequence) we may get σ1-positive/negative word.

Tetsuya Ito Braid calculus Aug , 2014 22 / 64 Handle reduction

Deﬁnition A permitted handle of a braid word w is a subword of the form

±1 ε ε ··· ε ∓1 h = σ1 V0σ2V1σ2 σ2Vk σ1 ∈ {± } ±1 ±1 where ε 1 and Vi is a word containing no σ1 , σ2 .

Not permitted Permitted

inner handle same sign

Tetsuya Ito Braid calculus Aug , 2014 23 / 64 Handle reduction

Deﬁnition The handle reduction of a permitted handle h in a braid word w is replacement ±1 ε ε ··· ε ∓1 h = σ1 V0σ2V1σ2 σ2Vk σ1 with ∓1 ε ±1 ··· ∓1 ε ±1 red(h) = V0(σ2 σ1σ2 )V1 (σ2 σ1σ2 )Vk

permitted handle handle reduction

Tetsuya Ito Braid calculus Aug , 2014 24 / 64 ▶ Handle reduction may create new handles (and the length of words may increase) – so it is unclear whether handle reduction makes braid in a better form. Are we approaching σ-positive/negative word ? ▶ Nevertheless, handle reduction eventually yields a σ-positive or σ-negative word:

Theorem (Dehornoy ’97) For a given n-braid word of length ℓ, after at most 2n4ℓ (exponential) times of handle reductions, we arrive at a σ-positive or σ-negative word.

The proof is not so simple, because we have no good notion of complexity which decrease by applying handle reduction.

Handle reduction

▶ A Handle reduction converts non σ-positive/negative subword h into a σ-positive/negative subword.

Tetsuya Ito Braid calculus Aug , 2014 25 / 64 ▶ Nevertheless, handle reduction eventually yields a σ-positive or σ-negative word:

Theorem (Dehornoy ’97) For a given n-braid word of length ℓ, after at most 2n4ℓ (exponential) times of handle reductions, we arrive at a σ-positive or σ-negative word.

The proof is not so simple, because we have no good notion of complexity which decrease by applying handle reduction.

Handle reduction

▶ A Handle reduction converts non σ-positive/negative subword h into a σ-positive/negative subword. ▶ Handle reduction may create new handles (and the length of words may increase) – so it is unclear whether handle reduction makes braid in a better form. Are we approaching σ-positive/negative word ?

Tetsuya Ito Braid calculus Aug , 2014 25 / 64 Theorem (Dehornoy ’97) For a given n-braid word of length ℓ, after at most 2n4ℓ (exponential) times of handle reductions, we arrive at a σ-positive or σ-negative word.

The proof is not so simple, because we have no good notion of complexity which decrease by applying handle reduction.

Handle reduction

Tetsuya Ito Braid calculus Aug , 2014 25 / 64 Handle reduction

Theorem (Dehornoy ’97) For a given n-braid word of length ℓ, after at most 2n4ℓ (exponential) times of handle reductions, we arrive at a σ-positive or σ-negative word.

The proof is not so simple, because we have no good notion of complexity which decrease by applying handle reduction.

Tetsuya Ito Braid calculus Aug , 2014 25 / 64 Find a handle:

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 . Handle reduction (get longer word!):

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Example of handle reduction

Let us use handle reduction to ﬁnd σ-positive or σ-negative word for

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 26 / 64 Handle reduction (get longer word!):

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Example of handle reduction

Let us use handle reduction to ﬁnd σ-positive or σ-negative word for

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 .

Find a handle:

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 26 / 64 Example of handle reduction

Let us use handle reduction to ﬁnd σ-positive or σ-negative word for

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 .

Find a handle:

−1 −1 −1 −1 −1 −1 −1 σ1σ2σ3σ2σ1 σ2 σ1 σ2 σ3 σ2 σ1 . Handle reduction (get longer word!):

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 26 / 64 ··· −1 By looking for the pattern σ1 σ1 , we ﬁnd:

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

This handle is not permitted – we look for inner handles

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

and do handle reduction (this is just a cancellation):

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 .

Example of handle reduction

Let us search next handle for

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64 This handle is not permitted – we look for inner handles

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

and do handle reduction (this is just a cancellation):

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 .

Example of handle reduction

Let us search next handle for

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 . ··· −1 By looking for the pattern σ1 σ1 , we ﬁnd:

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64 and do handle reduction (this is just a cancellation):

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 .

Example of handle reduction

Let us search next handle for

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 . ··· −1 By looking for the pattern σ1 σ1 , we ﬁnd:

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

This handle is not permitted – we look for inner handles

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64 Example of handle reduction

Let us search next handle for

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 . ··· −1 By looking for the pattern σ1 σ1 , we ﬁnd:

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

This handle is not permitted – we look for inner handles

−1 −1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ2σ2 σ1 σ2 σ3 σ2 σ1 .

and do handle reduction (this is just a cancellation):

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 Example of handle reduction

Handel reduction again:

−1 −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ1σ1 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . Iterate similar procedure:

−1 −1 −1 −1 −1 −1 σ2 σ1σ2σ3σ2 σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 .

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ2σ3σ2 σ3 σ2 σ1 . −1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

−1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 . −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

−1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 . −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 . −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

−1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 . −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 . −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

−1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 . −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 . −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

−1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 . −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 . −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

−1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 . −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 . −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

−1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 . −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

−1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 Example of handle reduction

−1 −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ3σ3 σ2 σ1 . −1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 .

−1 −1 −1 −1 −1 σ2 σ1σ3 σ3 σ2σ2 σ1 . −1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 .

−1 −1 −1 −1 σ2 σ1σ3 σ3 σ1 . −1 −1 −1 σ2 σ3 σ3 . We eventually ﬁnd σ-negative word (so, word without handles).

Tetsuya Ito Braid calculus Aug , 2014 29 / 64 Note that in the previous theorem only gives an exponential upper bound 24nℓ. This suggests our current understanding of handle reduction is very poor .

Handle reduction

Experimental fact Among known algorithms to compute the Dehornoy ordering, a handle reduction method is the best: ▶ It is easy to do (even by hands and to implement computor program). ▶ It converges in linear time (and will produce a short σ-positive/negative representatives).

Tetsuya Ito Braid calculus Aug , 2014 30 / 64 Handle reduction

Note that in the previous theorem only gives an exponential upper bound 24nℓ. This suggests our current understanding of handle reduction is very poor .

Tetsuya Ito Braid calculus Aug , 2014 30 / 64 Handle reduction

Question 1. Prove handle reduction converges very fast (conjectually in linear time, but polynomial time bound is still interesting) 2. Give a topological/geometric prospect of handle reduction. What is handle “reduction” reducing ? 3. Generalize a theory of handle reduction technique for other groups. (Note: handle reduction can be seen as standard reducing operation xx−1 7→ ε, which is basic in the free group. There might be a good notion and properties of “handle-reduced” words in more general group.)

A handle reduction seems to reﬂect unknown combinatorics and prospects in braid groups...

Tetsuya Ito Braid calculus Aug , 2014 31 / 64 II-2 Application to (contact) topology (1): Knot theory

Tetsuya Ito Braid calculus Aug , 2014 32 / 64 ▶ Set-theory (distribuitivive operations on sets) ▶ Surface triangulation (Dynnikov coordinate, Mosher’s normal form of MCG) ▶ Possibly more unknown prospects...... Moreover, in a theory of left-ordering of groups the Dehornoy ordering is a source of various important examples.

The Dehornoy ordering

Tetsuya Ito Braid calculus Aug , 2014 33 / 64 Moreover, in a theory of left-ordering of groups the Dehornoy ordering is a source of various important examples.

The Dehornoy ordering

Tetsuya Ito Braid calculus Aug , 2014 33 / 64 The Dehornoy ordering

The Dehornoy ordering

Tetsuya Ito Braid calculus Aug , 2014 33 / 64 Surprisingly, this speculation is true, and Conclusion If K is a closure of a braid β which is suﬃciently complicated (with respect to

The Dehornoy ordering

Natural question is: Question Can we use Dehornoy ordering to study topology/geometry ?

Naive speculation

The Dehornoy ordering knots and links, for example) arising from braids.

Tetsuya Ito Braid calculus Aug , 2014 34 / 64 The Dehornoy ordering

Natural question is: Question Can we use Dehornoy ordering to study topology/geometry ?

Naive speculation

The Dehornoy ordering

Surprisingly, this speculation is true, and Conclusion If K is a closure of a braid β which is suﬃciently complicated (with respect to

Tetsuya Ito Braid calculus Aug , 2014 34 / 64 Lemma

1. The Dehornoy ﬂoor map [ ]D : Bn → Z is a quasi-morphism of defect one: |[αβ]D − [α]D − [β]D | ≤ 1

2. If α and β are conjugate, |[α]D − [β]D | ≤ 1.

The Dehornoy ﬂoor of braids

Deﬁnition

The Dehornoy ﬂoor of braid β is an integer [β]D satisfying

2[βD ] 2[βD ]+2 ∆ ≤D β

The Dehornoy ﬂoor is regarded as a numerical complecity of braids measured by the Dehornoy ordering.

Tetsuya Ito Braid calculus Aug , 2014 35 / 64 The Dehornoy ﬂoor of braids

Deﬁnition

The Dehornoy ﬂoor of braid β is an integer [β]D satisfying

2[βD ] 2[βD ]+2 ∆ ≤D β

The Dehornoy ﬂoor is regarded as a numerical complecity of braids measured by the Dehornoy ordering. Lemma

1. The Dehornoy ﬂoor map [ ]D : Bn → Z is a quasi-morphism of defect one: |[αβ]D − [α]D − [β]D | ≤ 1

2. If α and β are conjugate, |[α]D − [β]D | ≤ 1.

Tetsuya Ito Braid calculus Aug , 2014 35 / 64 Proof: ±1 Assume β is conjugate to ασn−1, so is conjugate to ′ ±1 −1 { ±1 } ±1 β = ∆(ασn−1)∆ = word over σ2 , . . . , σn−1 σ1 . Then,

±2 ′ ±1 ±2 ±1 ∆ β = {word over σ , . . . , σ − }· (∆ σ ) 2 n 1 | {z 1 }

σ1−positive/negative So −2 ′ 2 ∆

The Dehornoy ﬂoor of braids Proposition If the closure of an n-braid β admits destabilization (i.e. β is conjugate to ±1 ∈ | | ≤ ασn for α Bn−1), then [β]D 1.

Tetsuya Ito Braid calculus Aug , 2014 36 / 64 The Dehornoy ﬂoor of braids Proposition If the closure of an n-braid β admits destabilization (i.e. β is conjugate to ±1 ∈ | | ≤ ασn for α Bn−1), then [β]D 1.

Proof: ±1 Assume β is conjugate to ασn−1, so is conjugate to ′ ±1 −1 { ±1 } ±1 β = ∆(ασn−1)∆ = word over σ2 , . . . , σn−1 σ1 . Then,

±2 ′ ±1 ±2 ±1 ∆ β = {word over σ , . . . , σ − }· (∆ σ ) 2 n 1 | {z 1 }

σ1−positive/negative So −2 ′ 2 ∆

Tetsuya Ito Braid calculus Aug , 2014 36 / 64 The Dehornoy ﬂoor of braids By the same argument, Proposition

1. If the closure of an n-braid β admits exchange move then |[β]D | ≤ 1.

2. If the closure of an n-braid β admits ﬂype then |[β]D | ≤ 2.

AB AB

Exchange Move

A A

C C B B Flype

Tetsuya Ito Braid calculus Aug , 2014 37 / 64 The Dehornoy ﬂoor of braids These “template moves” geometrically appears in the theory of Birman-Menasco’s braid foliation theory (cf. LaFountain’s lecture) and the Dehornoy ordering is related to the braid foliation. Key Proposition L = βb:Closed braid F ∈ S3 − L: Seifert/incompressible closed surface. If the braid foliation of F has a positive vertex v with p positive saddles

and n negative saddles around v, then −n ≤ [β]D ≤ p.

Ô Ó×iØiÚe ×addÐe×

ÒegaØiÚe ×addÐe×

µ ¬ ℄ D

Tetsuya Ito Braid calculus Aug , 2014 38 / 64 Dehornoy ﬂoor and knot theory By using the braid foliation theory, we have several close connections between the Dehornoy’s ordering (recall it is deﬁned by algebraic way !!) and knot theory: Proposition (Malyutin-Netsvetaev’04, I.) b 1. If |[β]D | > 1, then β is prime, non-split, non-trivial link. 2. For n ∈ {2, 3,..., } there exists a number r(n) ∈ Z such that: The closure of two n-braids α and β with |[α]D |, |[β]D | ≥ r(n) represent the same link if and only if α and β are conjugate. (Moreover, the braid index of αb = n).

Surprising consequence If the Dehornoy ﬂoor is suﬃciently large,

Algebraic link problem = conjugacy problem of braids!!

Tetsuya Ito Braid calculus Aug , 2014 39 / 64 Dehornoy ﬂoor and knot theory

More direct connections for Dehornoy ordering and knots: Theorem (I, ’12) b b 1. Let β ∈ Bn. If g(β) the genus of a knot β, 4g(K) 2 3 |[β] | ≤ − + ≤ g(K) + 1. D n + 2 n + 2 2 Thus, a complicated braid (with respect to the Dehornoy ordering) yields a complicated knot (with respect to topology – Thurston norm)

2. Assume that |[β]D | ≥ 2. Then,    torus knot  periodic b ⇐⇒ β is a  satellite knot β is  reducible hyperbolic knot pseudo-Anosov

Tetsuya Ito Braid calculus Aug , 2014 40 / 64 Further application: quantum invariants

Recall the deﬁnitions of quantum invariants: Uq(g): Quantum enveloping algebra of semi-simple Lie algebra g V : Uq(g)-module(s)

Surgery {Braids}Closure / {(Oriented) Links } / {Closed 3-manifolds}

Quantum Quantum Quantum ρ V representation invariant invariant √ 2π −1 ′′ q=e N GL(V ⊗n) “Trace / C[q, q−1] / C Take linear sums

ρV : Bn → GL(V ) is called quantum representation.

Tetsuya Ito Braid calculus Aug , 2014 41 / 64 From the construction of quantum invariants, we have Observations (Bigelow)

If an n-braid α ∈ KerρV , then for any β ∈ Bn c b QV (αβ) = QV (β)

⇒ Quantum representation ρV is not faithful, then QV is not strong – it fails to detect the unknot. Is it true? The link αβc may be the same as βb...

Big open problem in knot theory

Open problem Which quantum invariants detect the unknot ? Does Jones polynomial (“the simplest” quantum invariant) detect the unknot ?

Tetsuya Ito Braid calculus Aug , 2014 42 / 64 Big open problem in knot theory

Open problem Which quantum invariants detect the unknot ? Does Jones polynomial (“the simplest” quantum invariant) detect the unknot ? From the construction of quantum invariants, we have Observations (Bigelow)

If an n-braid α ∈ KerρV , then for any β ∈ Bn c b QV (αβ) = QV (β)

⇒ Quantum representation ρV is not faithful, then QV is not strong – it fails to detect the unknot. Is it true? The link αβc may be the same as βb...

Tetsuya Ito Braid calculus Aug , 2014 42 / 64 Closed braids via normal subgroups

Bigelow’s speculation is true: Theorem (I.)

Let N be the non-trivial, non-central normal subgroup of Bn. Then for any β ∈ Bn, the set of knots (and links)

{αβc | α ∈ N}

contains inﬁnitely may distinct (hyperbolic) knots. (i.e. normal subgroup of Bn produces inﬁnitely many knots.)

It sounds “obvious”, but how to prove ? ▶ We cannot use (easy-to-calculate) invariant to distinguish knots !!! ▶ We do not know an element in N explicitly.

Tetsuya Ito Braid calculus Aug , 2014 43 / 64 Consequences

One can attack faithfulness of knot invariants via braid group representations: Corollary (I.) → ⊗n 1. If quantum representations ρi : Bn GL(Vi )(i = 1,..., k) are not faithful, for any knot type K, there exists inﬁnitely many mutually diﬀerent, (hyperbolic) knot K1, K2,... such that ∗ QVi (K) = QVi (K∗)( = 1, 2,...) for all i = 1,..., k.

2. (Bigelow) If the 4-strand (reduced) Burau representation

±1 ρ4 : B4 → GL(3, Z[q ])

is not faithful, then there exists a non-trivial knot with trivial Jones polynomial.

Tetsuya Ito Braid calculus Aug , 2014 44 / 64 Proof of Theorem Surprisingly, theorem is a consequence of a purely algebraic statement for the Dehornoy ordering. Theorem′ (I.)

A non-trivial normal subgroup N of Bn is unbounded with respect to the Dehornoy ordering

Theorem′ ⇒ Theorem ▶ If N is unbounded, so is the set {βα | α ∈ N} for any β. (Moreover, it contains infinitely many pseudo-Anosov elements). ▶ Inequality of the Dehornoy floor and knot genus (previous theorem) shows the set of knots {βαc | α ∈ N} is infinite (because it contains arbitrary large genus knot.)

Tetsuya Ito Braid calculus Aug , 2014 45 / 64 II-4: Application (2): FDTC and contact geometry

Tetsuya Ito Braid calculus Aug , 2014 46 / 64 1. Periodic case: N M ··· Take N > 0 so that ϕ = TC (Dehn twsits along other boundaries). Then, M c(ϕ, C) = N 2πM (we regard ϕ is rotation by N near C)

Fractional Dehn twist coeﬃcient

S: surface with non-empty boundary C ⊂ ∂S: connected component of ∂S. Using Nielsen-Thurston theory, Honda-Kazez-Matićdefined the fractional Dehn twist coefficients (FDTC) (with respect to C) c(ϕ, C) ∈ Q:

Tetsuya Ito Braid calculus Aug , 2014 47 / 64 Fractional Dehn twist coeﬃcient

1. Periodic case: N M ··· Take N > 0 so that ϕ = TC (Dehn twsits along other boundaries). Then, M c(ϕ, C) = N 2πM (we regard ϕ is rotation by N near C)

Tetsuya Ito Braid calculus Aug , 2014 47 / 64 Fractional Dehn twist coeﬃcient

2. Pseudo-Anosov case: Consider a pseudo-Anosov homormophism representative. Using the singular leaves of its invariant foliation near C, in the neighborhood of C 2πM we identify ϕ with the rotation by N , and deﬁne M c(ϕ, C) = N

3. Reducible case: Consider irreducible component of ϕ containing C.

Tetsuya Ito Braid calculus Aug , 2014 48 / 64 Theorem (I-Kawamuro)

c(ϕ, C) = τ ◦ Θ(ϕ).

Alternative deﬁnition of FDTC Let us use a Nielsen-Thurston map

Θ: MCG(S) → Homeo+(R).

TC is central and we may normalize Θ so that Θ(TC ): x 7→ x + 1: i.e., ^ Θ: MCG(S) → Homeo+(S1)

^ (Here Homeo+(S1) = {ef : R → R | f : R/Z = S1 → S1}). Consider the translation number ^ [Θ(ψN )](0) − 0 τ : Homeo+(S1) → R, τ(ψ) = lim . N→∞ N

Tetsuya Ito Braid calculus Aug , 2014 49 / 64 Alternative deﬁnition of FDTC Let us use a Nielsen-Thurston map

Θ: MCG(S) → Homeo+(R).

TC is central and we may normalize Θ so that Θ(TC ): x 7→ x + 1: i.e., ^ Θ: MCG(S) → Homeo+(S1)

^ (Here Homeo+(S1) = {ef : R → R | f : R/Z = S1 → S1}). Consider the translation number ^ [Θ(ψN )](0) − 0 τ : Homeo+(S1) → R, τ(ψ) = lim . N→∞ N

Theorem (I-Kawamuro)

c(ϕ, C) = τ ◦ Θ(ϕ).

Tetsuya Ito Braid calculus Aug , 2014 49 / 64 Relation to the Dehornoy ﬂoor

Let us consider a normalized Nielsen-Thurston map for braids,

^+ 1 Θ: Bn → Homeo (S )

Θ(∆2): x 7→ x + 1. Moreover, we may further arrange so that

Then, the translation number is nothing but the “stable” Dehornoy ﬂoor: Theorem (I-Kawamuro) [βN ] c(ϕ, C) = lim D N→∞ N

Tetsuya Ito Braid calculus Aug , 2014 50 / 64 ▶ Fast computation of FDTC without knowing Nielsen-Thurston type. (For the braid group, we have fast (conjectured to be linear time) algorithm to compute the Dehornoy ordering. ) ▶ Relationship between Nielsen-Thurston ordering of Mapping class groups and contact 3-manifolds.

Application to contact geometry

Summary FDTC = generalization of the Dehornoy ﬂoor

In particular, we have:

Tetsuya Ito Braid calculus Aug , 2014 51 / 64 ▶ Relationship between Nielsen-Thurston ordering of Mapping class groups and contact 3-manifolds.

Application to contact geometry

Summary FDTC = generalization of the Dehornoy ﬂoor

Tetsuya Ito Braid calculus Aug , 2014 51 / 64 Application to contact geometry

Summary FDTC = generalization of the Dehornoy ﬂoor

In particular, we have: ▶ Fast computation of FDTC without knowing Nielsen-Thurston type. (For the braid group, we have fast (conjectured to be linear time) algorithm to compute the Dehornoy ordering. ) ▶ Relationship between Nielsen-Thurston ordering of Mapping class groups and contact 3-manifolds.

Tetsuya Ito Braid calculus Aug , 2014 51 / 64 Application to contact geometry

Viewing FDTC as generalization of Dehornoy ﬂoor (and the theory of Open book foliation) allows us to generalize theorem for Dehornoy ordering and knot theory for FDTC and (contact) 3-manifolds. Theorem (I-Kawamuro) Let (S, ϕ) be an open book decomposition of (contact) 3-manifold (M, ξ).

1. If |c(ϕ, C)| ≥ 4 for all boundary of S, then    Seifert-ﬁbered manifold  periodic ⇐⇒ M is a  toroidal manifold ϕ is  reducible hyperbolic manifold pseudo-Anosov

2. If S is planar and c(ϕ, C) > 1 for all boundary of S, then ξ is a tight contact structure on M.

Tetsuya Ito Braid calculus Aug , 2014 52 / 64 II-5: The Dehornoy ordering and Garside theory

Tetsuya Ito Braid calculus Aug , 2014 53 / 64 At ﬁrst glance: ▶ There seems to be little connection between Garside normal form and Dehornoy ordering – normal form is far from σ-positive word. ▶ However,

Garside theory technique to compute Dehornoy’s ordering: Alternating normal form

Question Can we use Garside structure to study/compute normal form ?

Tetsuya Ito Braid calculus Aug , 2014 54 / 64 Natural speculation + ≼ For Bn , using normal forms we can extend to get the Dehornoy ordering

Garside theory technique to compute Dehornoy’s ordering: Alternating normal form

Question Can we use Garside structure to study/compute normal form ?

At ﬁrst glance: ▶ There seems to be little connection between Garside normal form and Dehornoy ordering – normal form is far from σ-positive word. ▶ However,

Tetsuya Ito Braid calculus Aug , 2014 54 / 64 Garside theory technique to compute Dehornoy’s ordering: Alternating normal form

Question Can we use Garside structure to study/compute normal form ?

At ﬁrst glance: ▶ There seems to be little connection between Garside normal form and Dehornoy ordering – normal form is far from σ-positive word. ▶ However,

Tetsuya Ito Braid calculus Aug , 2014 54 / 64 Geometric speculation From the curve diagram interpretation...

(Braids without σ1)

(Braids without σn−1)(Braids without σ1)

Chasing the patterns of appearance of σ1 and σn−1 can estimate the Dehornoy ordering.

Tetsuya Ito Braid calculus Aug , 2014 55 / 64 Garside theory technique to compute Dehornoy’s ordering: Alternating normal form

To capture the patterns of σ1 and σn−1, we use Garside structure idea: Let us deﬁne { { } ⊂ + A = Positive words over σ1, . . . , σn−2 Bn { } ⊂ + B = Positive words over σ2, dots, σn−1 Bn

and { ∆A = (σ1σ2 ··· σn−2)(σ1σ2 ··· σn−3) ··· (σ1σ2)(σ1) ∈ A

∆B = (σ2σ3 ··· σn−1)(σ2σ3 ··· σn−2) ··· (σ2σ3)(σ2) ∈ B

+ (Both A and B are isomorphic to Bn−1.)

Tetsuya Ito Braid calculus Aug , 2014 56 / 64 Garside theory technique to compute Dehornoy’s ordering: Alternating normal form

Recall that in the normal form, we decompose β as a product of simple elements by repeatedly computing β ∧ ∆. WE replace the role of ∆ by A and B. Deﬁnition ∈ + For β Bn , deﬁne ∧ ∧ N ∈ β A = max β ∆A A, ≼,N>0 = max{α ∈ A | βα−1 ∈ A}. ≼

β ∧ B is deﬁned similarly.

Tetsuya Ito Braid calculus Aug , 2014 57 / 64 Example 3 For β = (σ1σ2) = σ1σ2σ1σ2σ1σ2, A · 2 · · 2 (β) = σ1 σ2 σ1 σ2

Alternating normal form

Deﬁnition ∈ + For β Bn , the alternating decomposition is a factorizatino of β

A(β) = bk ak ··· b1a1b0

where ai ∈ A, bi ∈ B is deﬁned by:  b0 = β ∧ B −1 a = β(b − ··· b a b ) ∧ A  i i 1 1 1 0 −1 bi = β(ai ··· b1a1b0) ∧ B

Tetsuya Ito Braid calculus Aug , 2014 58 / 64 Alternating normal form

Deﬁnition ∈ + For β Bn , the alternating decomposition is a factorizatino of β

A(β) = bk ak ··· b1a1b0

where ai ∈ A, bi ∈ B is deﬁned by:  b0 = β ∧ B −1 a = β(b − ··· b a b ) ∧ A  i i 1 1 1 0 −1 bi = β(ai ··· b1a1b0) ∧ B

Example 3 For β = (σ1σ2) = σ1σ2σ1σ2σ1σ2, A · 2 · · 2 (β) = σ1 σ2 σ1 σ2

Tetsuya Ito Braid calculus Aug , 2014 58 / 64 Alternating normal form Note we have an injective homomorphism → ∼ + ⊂ + −1 Φ: B A = Bn−1 Bn , Φ(β) = ∆β∆

Theorem (Dehornoy ’09, I. ’10) For two positive braids α and β, let { A(α) = bk ak ··· b1a1b0

A(β) = Bk′ Ak′ ··· B1A1B0

be their alternating decomposition. Then   ′ k < k or,  k = k′ and ∃i such that α < β ⇐⇒ D  ··· Bk = bk , Ak = ak , Ai+1 = ai+1.  Φ(Bi )

Tetsuya Ito Braid calculus Aug , 2014 59 / 64 Alternating normal form

Theorem (Dehornoy ’09, I. ’10) That is, sequence of (n − 1) braids coming from alternating decomposition of β (,..., Φ(Bk′ ), Ak′ ,..., Φ(B1), A1, Φ(B0)) is larger than the sequence from α

(,..., Φ(bk ), ak ,..., Φ(b1), a0, Φ(b0))

with respect to the lexicographical ordering based on the Dehornoy ordering

Thus, by using Garside theory method, we can reduce the computation of the Dehornoy ordering of Bn to the computation in Bn−1.

Tetsuya Ito Braid calculus Aug , 2014 60 / 64 Iteration of alternating decomposition provides nice enumeration of positive braids with respect to

Application

Like normal forms, alternating decomposition is easy to calculate: Proposition (Dehornoy) The alternating decomposition of positive braid of length ℓ is computed in time O(ℓ2). In particular, for given (not necessarily positive) braid β of 2 length ℓ, whether 1

Tetsuya Ito Braid calculus Aug , 2014 61 / 64 Application

Iteration of alternating decomposition provides nice enumeration of positive braids with respect to

Tetsuya Ito Braid calculus Aug , 2014 61 / 64 Remarks 1. Similar arguments apply for the dual Garside structure (Fromentin, I.)

In the classical case, we used

+ + −1 A = Bn−1 and B = ∆(Bn−1)∆ . In the dual case, we use

+∗ +∗ −1 2 +∗ −2 A1 = Bn−1, A2 = δBn−1δ , A3 = δ Bn−1δ ,... 2. The Dehornoy ordering is a speicial one of the Nielsen-Thurston type ordering. Using the variant of alternating normal form, we have a similar result for a special class of Nielsen-Thurston type ordering called of ﬁnite type.

Note: For Nielsen-Thurston type orderings, we can not use σ-positive words – now, the Garside structure provides algebraic, combinatorial description of geometrically deﬁned orderings.

Tetsuya Ito Braid calculus Aug , 2014 62 / 64 Further readings

For basics of Garside normal forms, Section 1 of ▶ J. Birman, V. Gebhardt, and J. Gonz´alez-Meneses,Conjugacy in Garside group I: cycling, powers and rigidity, Groups Geom. Dyn 1 (2007) 221-279. or, Section 5 of ▶ J. Birman, and T. Brendle, Braids: A survey, Handbook of knot theory, 19–103. contains a nice and concise overview.

Tetsuya Ito Braid calculus Aug , 2014 63 / 64 Further readings

For the basics of the Dehornoy ordering, ▶ P. Dehornoy, I.Dynnikov, D.Rolfsen and B.Wiest, Ordering Braids, Mathematical Surveys and Monographs 148, Amer. Math. Soc. 2008. Also, a survey ▶ P. Dehornoy Braid order, sets, and knots. Introductory lectures on knot theory, 77-96, Ser. Knots Everything, 46, World Sci. Publ., Hackensack, NJ, 2012. contains brief explanation of applications to knot theory.

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