Garside structure and Dehornoy ordering of braid groups for - - PowerPoint PPT Presentation

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

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Part II Dehornoy’s ordering

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II-1: Dehornoy’s ordering: definition

Tetsuya Ito Braid calculus Aug , 2014 4 / 64

σ-positive word

Definition

W : word over {σ±1

1 , . . . , σ±1 n−1}

W is      σi − positive ⇐ ⇒ W contains at least one σi does not contain σ±1

1 , . . . , σ±1 i−1, σ−1 i

. σi − negative ⇐ ⇒ W contains at least one σi −1 does not contain σ±1

1 , . . . , σ±1 i−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

σ-positive word

Examples:

▶ W = σ2σ−1 3 : σ2-positive word

Tetsuya Ito Braid calculus Aug , 2014 6 / 64

σ-positive word

Examples:

▶ W = σ2σ−1 3 : σ2-positive word ▶ W = σ1σ2σ−1 1 : Neither σ-positive nor σ-negative word.

Tetsuya Ito Braid calculus Aug , 2014 6 / 64

σ-positive word

Examples:

▶ W = σ2σ−1 3 : σ2-positive word ▶ W = σ1σ2σ−1 1 : Neither σ-positive nor σ-negative word. ▶ As a braid, β = σ1σ2σ−1 1

is σ-positive: σ1σ2σ−1

1

= σ−1

2 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: Is a braid σ1σ3

2σ−2 1 σ−2 2 σ1σ2σ2 1σ−7 2

σ-positive ?

Tetsuya Ito Braid calculus Aug , 2014 6 / 64

Dehornoy’s ordering: Algebraic definition

Theorem-Definition (Denornoy ’94)

Define the relation <D of Bn by α <D β ⇐ ⇒ α−1β is σ-positive as a braid. Then <D is a left ordering of Bn (Denornoy ordering): that is, <D is a total ordering and is left-invariant relation: α <D β = ⇒ γα <D γβ (∀α, β, γ ∈ Bn)

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

Consequence of orderability

From the simple fact that Bn admits a left-ordering <D (without knowing how we defined it), we can deduce several algebraic properties of Bn

Tetsuya Ito Braid calculus Aug , 2014 8 / 64

Consequence of orderability

From the simple fact that Bn admits a left-ordering <D (without knowing how we defined it), we can deduce several algebraic properties of Bn

• 1. Bn is torsion-free:

For non trivial β, either 1 <D β or 1 >D β. Then { 1 <D β ⇒ 1 <D β <D β2 <D · · · <D βi <D · · · 1 >D β ⇒ 1 >D β >D β2 >D · · · >D βi >D · · ·

Tetsuya Ito Braid calculus Aug , 2014 8 / 64

Consequence of orderability

From the simple fact that Bn admits a left-ordering <D (without knowing how we defined it), we can deduce several algebraic properties of Bn

• 1. Bn is torsion-free:

For non trivial β, either 1 <D β or 1 >D β. Then { 1 <D β ⇒ 1 <D β <D β2 <D · · · <D βi <D · · · 1 >D β ⇒ 1 >D β >D β2 >D · · · >D βi >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

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

• 1. 1 <D σn−1 <D · · · <D σ2 <D σ1.

Tetsuya Ito Braid calculus Aug , 2014 9 / 64

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

• 1. 1 <D σn−1 <D · · · <D σ2 <D σ1.
• 2. σk

2 <D σ1 for any k > 0.

Tetsuya Ito Braid calculus Aug , 2014 9 / 64

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

• 1. 1 <D σn−1 <D · · · <D σ2 <D σ1.
• 2. σk

2 <D σ1 for any k > 0.

• 3. 1 <D σ1σ−1

2

Tetsuya Ito Braid calculus Aug , 2014 9 / 64

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

• 1. 1 <D σn−1 <D · · · <D σ2 <D σ1.
• 2. σk

2 <D σ1 for any k > 0.

• 3. 1 <D σ1σ−1

2

, but 1 = (σ1σ2σ1)1(σ1σ2σ1)−1 >D (σ1σ2σ1)(σ1σ−1

2 )(σ1σ2σ1)−1 = σ2σ−1 1

so, the relation <D is not preserved under conjugacy.

Tetsuya Ito Braid calculus Aug , 2014 9 / 64

Dehornoy’s ordering: example

Let us look some properties of Dehornoy’s ordering

• 1. 1 <D σn−1 <D · · · <D σ2 <D σ1.
• 2. σk

2 <D σ1 for any k > 0.

• 3. 1 <D σ1σ−1

2

, but 1 = (σ1σ2σ1)1(σ1σ2σ1)−1 >D (σ1σ2σ1)(σ1σ−1

2 )(σ1σ2σ1)−1 = σ2σ−1 1

so, the relation <D is not preserved under conjugacy.

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

particular: For any β ∈ Bn, there exists N ∈ Z such that β <D ∆N. (Compare Property 2. above)

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Property S (Subword Property)

Theorem (Property S, Lavor ’96)

For any β ∈ Bn, 1 <D βσiβ−1. Property S has several consequences:

Corollary

• 1. αβ <D ασiβ for any α, β ∈ Bn.
• 2. A band generator is <D-positive: 1 <D ai,j.
• 3. <D is an extension of partial ordering in the classical Garside

structure: α ≼ β ⇒ α <D β

• 4. <D is an extension of partial ordering in the dual Garside structure:

α ≼∗ β ⇒ α <D β

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Well-orderedness

An important consequence of the Property S is:

Theorem (Lavar, Burckel, Dehornoy, Fromentin, I)

• 1. The restriction of <D on B+

n is a well-ordering (every non-empty set

admits the <D-minimal element), and the ordinal of the ordered set (B+

n , <D) is ωωn−2.

• 2. The restriction of <D on B+∗

n

is a well-ordering and the ordinal of the

• rdered set (B+∗

n , <D) is ωωn−2.

(Note: B+

n ⊂ B+∗ n .)

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 <D on B+

n is a well-ordering (every non-empty set

admits the <D-minimal element), and the ordinal of the ordered set (B+

n , <D) is ωωn−2.

• 2. The restriction of <D on B+∗

n

is a well-ordering and the ordinal of the

• rdered set (B+∗

n , <D) is ωωn−2.

(Note: B+

n ⊂ B+∗ n .)

An existence of <D-minimal element seems to be useful, but at this moment, no application is known.

Open Problem

Can we compute the <D-minimum element of the set of positive conjugates {αβα−1 | α ∈ Bn} ∩ B+

n of β ?

Tetsuya Ito Braid calculus Aug , 2014 11 / 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 ?

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: For a given n-braid β with length ℓ (with respect to {σ±1

1 , . . . , σ±1 n−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)

α <D β ⇐ ⇒ β(Γ)“moves the left side” of α(Γ) when we put β(Γ) and α(Γ) intersect minimally

• 1

• 2

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Dehornoy’s ordering: geometric view (2)

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

2 , . . . , σ±1 n−1})

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

• 1

• 1

• 1

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

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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:

• 1

• 1

• 1

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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

Dehornoy’s ordering: geometric view (3)

Application

Geometric point of view yields:

▶ A simple proof of Property S. ▶ Algorithm to determine 1 <D β or not

(just try to write curve diagram !)

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 <D β or not

(just try to write curve diagram !)

▶ Algorithm to find σ-positive representative word of β if 1 <D β

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 <D β or not

(just try to write curve diagram !)

▶ Algorithm to find σ-positive representative word of β if 1 <D β

Important prospect

A reasonable procedure of simplifying curve diagrams yields a connection

• f 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...)

Tetsuya Ito Braid calculus Aug , 2014 16 / 64

Dehornoy’s ordering: More schematic geometric view

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

Dehornoy’s ordering: More schematic geometric view

We want to remove “up to isotopy” in curve diagram definition of <D. 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,

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 <D. 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. ▶ In the universal covering of Dn ⊂ H2, 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)

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Nielsen-Thurston type ordering

Using hyperbolic geometry construction, we can generalize the Dehornoy

• rdering for mapping class group of surface with non-empty boundary:

S: Hyperbolic Surface with non-empty geodesic boundary H2 ⊃ S

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...)

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Nielsen-Thurston type ordering

Definition

Take an ordered, countable dense subset {x1, x2, . . .} of R. For ϕ, ψ ∈ MCG(S), define ϕ < ψ ⇐ ⇒ ∃j s.t. { [Θ(ϕ)] (xi) = [Θ(ψ)] (xi) i = 1, . . . , j − 1 [Θ(ϕ)] (xj) < [Θ(ψ)] (xj) This defines a left ordering of MCG(S), called the Nielsen-Thurston type

• rderings.

Remark

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

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II -2 Technique to compute Dehornoy

• rdering (handle reduction)

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Handle reduction

How to determine 1 <D β or not ?

Observation

σ1σ2σ−1

1

is not σ-positive word, but we can rewrite σ1σ2σ−1

1

= σ−1

2 σ1σ2 (σ1-positive word)

More generally, σ1σk

2σ−1 1

= σ−1

2 σk 1σ2

Tetsuya Ito Braid calculus Aug , 2014 22 / 64

Handle reduction

How to determine 1 <D β or not ?

Observation

σ1σ2σ−1

1

is not σ-positive word, but we can rewrite σ1σ2σ−1

1

= σ−1

2 σ1σ2 (σ1-positive word)

More generally, σ1σk

2σ−1 1

= σ−1

2 σk 1σ2

Idea: By modifying the word of the form σ1(σ1-free word)σ−1

1

(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 h = σ±1

1 V0σε 2V1σε 2 · · · σε 2Vkσ∓1 1

where ε ∈ {±1} and Vi is a word containing no σ±1

1 , σ±1 2 .

Not permitted Permitted

inner handle same sign

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Handle reduction

Definition

The handle reduction of a permitted handle h in a braid word w is replacement h = σ±1

1 V0σε 2V1σε 2 · · · σε 2Vkσ∓1 1

with red(h) = V0(σ∓1

2 σε 1σ±1 2 )V1 · · · (σ∓1 2 σε 1σ±1 2 )Vk

permitted handle handle reduction

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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

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

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 ?

▶ Nevertheless, handle reduction eventually yields a σ-positive or

σ-negative word:

Tetsuya Ito Braid calculus Aug , 2014 25 / 64

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 ?

▶ 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

• f 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

Example of handle reduction

Let us use handle reduction to find σ-positive or σ-negative word for σ1σ2σ3σ2σ−1

1 σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

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 σ1σ2σ3σ2σ−1

1 σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Find a handle: σ1σ2σ3σ2σ−1

1 σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

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 σ1σ2σ3σ2σ−1

1 σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Find a handle: σ1σ2σ3σ2σ−1

1 σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Handle reduction (get longer word!): σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 26 / 64

Example of handle reduction

Let us search next handle for σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64

Example of handle reduction

Let us search next handle for σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

By looking for the pattern σ1 · · · σ−1

1 , we find:

σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64

Example of handle reduction

Let us search next handle for σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

By looking for the pattern σ1 · · · σ−1

1 , we find:

σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

This handle is not permitted – we look for inner handles σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64

Example of handle reduction

Let us search next handle for σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

By looking for the pattern σ1 · · · σ−1

1 , we find:

σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

This handle is not permitted – we look for inner handles σ−1

2 σ1σ2σ3σ−1 2 σ1σ2σ−1 2 σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

and do handle reduction (this is just a cancellation): σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 27 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Iterate similar procedure:

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Iterate similar procedure: σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Iterate similar procedure: σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ2σ3σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Iterate similar procedure: σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ2σ3σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ2σ3σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

Handel reduction again: σ−1

2 σ1σ2σ3σ−1 2 σ1σ−1 1 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

Iterate similar procedure: σ−1

2 σ1σ2σ3σ−1 2 σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ2σ3σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ2σ3σ−1 2 σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 28 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ−1 1 .

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

Example of handle reduction

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ3σ−1 3 σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ2σ−1 2 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ−1 1 .

σ−1

2 σ1σ−1 3 σ−1 3 σ−1 1 .

σ−1

2 σ−1 3 σ−1 3 .

We eventually find σ-negative word (so, word without handles).

Tetsuya Ito Braid calculus Aug , 2014 29 / 64

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

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). 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 → 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

The Dehornoy ordering

The Dehornoy ordering <D is fundamental, but quite interesting object related to various aspects of the braid groups:

▶ Combinatorics (σ-positive words) ▶ Topology (Curve diagram) ▶ Geometry (Hyperbolic geometry, Nielsen-Thruston action)

Tetsuya Ito Braid calculus Aug , 2014 33 / 64

The Dehornoy ordering

The Dehornoy ordering <D is fundamental, but quite interesting object related to various aspects of the braid groups:

▶ Combinatorics (σ-positive words) ▶ Topology (Curve diagram) ▶ Geometry (Hyperbolic geometry, Nielsen-Thruston action) ▶ Set-theory (distribuitivive operations on sets) ▶ Surface triangulation (Dynnikov coordinate, Mosher’s normal form of

MCG)

▶ Possibly more unknown prospects......

Tetsuya Ito Braid calculus Aug , 2014 33 / 64

The Dehornoy ordering

The Dehornoy ordering <D is fundamental, but quite interesting object related to various aspects of the braid groups:

▶ Combinatorics (σ-positive words) ▶ Topology (Curve diagram) ▶ Geometry (Hyperbolic geometry, Nielsen-Thruston action) ▶ 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.

Tetsuya Ito Braid calculus Aug , 2014 33 / 64

The Dehornoy ordering

Natural question is:

Question

Can we use Dehornoy ordering to study topology/geometry ?

Naive speculation

The Dehornoy ordering <D can be seen as a complexity of braids. ⇒ <D may also be regarded as a complexity of geometric object (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 <D can be seen as a complexity of braids. ⇒ <D may also be regarded as a complexity of geometric object (knots and links, for example) arising from braids. Surprisingly, this speculation is true, and

Conclusion

If K is a closure of a braid β which is sufficiently complicated (with respect to <D), then property of K is directly read from β.

Tetsuya Ito Braid calculus Aug , 2014 34 / 64

The Dehornoy floor of braids

Definition

The Dehornoy floor of braid β is an integer [β]D satisfying ∆2[βD] ≤D β <D ∆2[βD]+2 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] ≤D β <D ∆2[βD]+2 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
• ne:

|[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

The Dehornoy floor of braids

Proposition

If the closure of an n-braid β admits destabilization (i.e. β is conjugate to ασ±1

n

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 ασ±1

n

for α ∈ Bn−1), then |[β]D| ≤ 1. Proof: Assume β is conjugate to ασ±1

n−1, so is conjugate to

β′ = ∆(ασ±1

n−1)∆−1 = {word over σ±1 2 , . . . , σn−1}σ±1 1 .

Then, ∆±2β′ = {word over σ±1

2 , . . . , σn−1} ·

(∆±2σ±1

1 )

• σ1−positive/negative

So ∆−2 <D β′ <D ∆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.

Exchange Move Flype

A B

A B C

A B

A B C

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 = β: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.

• sitive

• [

• 3

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.)

• 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 α = 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)

• 1. Let β ∈ Bn. If g(

β) the genus of a knot β, |[β]D| ≤ 4g(K) n + 2 − 2 n + 2 + 3 2 ≤ g(K) + 1. 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,
• β is a

   torus knot satellite knot hyperbolic knot ⇐ ⇒ β is    periodic reducible 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) {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

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

Tetsuya Ito Braid calculus Aug , 2014 41 / 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 ?

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 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 αβ may be the same as β...

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) { αβ | α ∈ N} contains infinitely may distinct (hyperbolic) knots. (i.e. normal subgroup

• f 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.)

• 1. If quantum representations ρi : Bn → GL(V ⊗n

i

) (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

ρ4 : B4 → GL(3, Z[q±1]) 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 <D: For any β ∈ Bn, there exists α ∈ N such that α−1 <D β <D α. 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 { βα | 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

Fractional Dehn twist coefficient

S: surface with non-empty boundary C ⊂ ∂S: connected component of ∂S. Using Nielsen-Thurston theory, Honda-Kazez-Mati´ c defined 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

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

• 1. Periodic case:

Take N > 0 so that ϕN = T M

C · · · (Dehn twsits along other boundaries).

Then, c(ϕ, C) = M N (we regard ϕ is rotation by 2πM

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 we identify ϕ with the rotation by 2πM

N , and define

c(ϕ, C) = M N

• 3. Reducible case:

Consider irreducible component of ϕ containing C.

Tetsuya Ito Braid calculus Aug , 2014 48 / 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 → x + 1: i.e., Θ : MCG(S) →

• Homeo+(S1)

(Here

• Homeo+(S1) = {

f : R → R | f : R/Z = S1 → S1}). Consider the translation number τ :

• Homeo+(S1) → R,

τ(ψ) = lim

N→∞

[Θ(ψN)](0) − 0 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 → x + 1: i.e., Θ : MCG(S) →

• Homeo+(S1)

(Here

• Homeo+(S1) = {

f : R → R | f : R/Z = S1 → S1}). Consider the translation number τ :

• Homeo+(S1) → R,

τ(ψ) = lim

N→∞

[Θ(ψN)](0) − 0 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, Θ : Bn →

• Homeo+(S1)

Θ(∆2) : x → x + 1. Moreover, we may further arrange so that α <D β ⇐ ⇒ [Θ(α)](0) < [Θ(β)](0). Then, the translation number is nothing but the “stable” Dehornoy floor:

Theorem (I-Kawamuro)

c(ϕ, C) = lim

N→∞

[βN]D N

Tetsuya Ito Braid calculus Aug , 2014 50 / 64

Application to contact geometry

Summary

FDTC = generalization of the Dehornoy floor In particular, we have:

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. )

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

• rdering 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

M is a    Seifert-fibered manifold toroidal manifold hyperbolic manifold ⇐ ⇒ ϕ is    periodic reducible 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

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

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, <D is an extension of subword ordering ≼ in Garside theory.

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, <D is an extension of subword ordering ≼ in Garside theory.

Natural speculation

For B+

n , using normal forms we can extend ≼ to get the Dehornoy

• rdering <D.

Tetsuya Ito Braid calculus Aug , 2014 54 / 64

Geometric speculation

From the curve diagram interpretation... (Braids without σ1) <D σ1 (Braids without σn−1)(Braids without σ1) <D σn−1 · · · σ2σ1 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} ⊂ B+

n

B = {Positive words over σ2, dots, σn−1} ⊂ B+

n

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 B+

n−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 ∈ B+

n , define

β ∧ A = max

≼,N>0 β ∧ ∆N A ∈ A,

= max

≼ {α ∈ A | βα−1 ∈ A}.

β ∧ B is defined similarly.

Tetsuya Ito Braid calculus Aug , 2014 57 / 64

Alternating normal form

Definition

For β ∈ B+

n , the alternating decomposition is a factorizatino of β

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

Tetsuya Ito Braid calculus Aug , 2014 58 / 64

Alternating normal form

Definition

For β ∈ B+

n , the alternating decomposition is a factorizatino of β

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

Example

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

2 · σ1 · σ2 2

Tetsuya Ito Braid calculus Aug , 2014 58 / 64

Alternating normal form

Note we have an injective homomorphism Φ : B → A ∼ = B+

n−1 ⊂ B+ n ,

Φ(β) = ∆β∆−1

Theorem (Dehornoy ’09, I. ’10)

For two positive braids α and β, let { A(α) = bkak · · · b1a1b0 A(β) = Bk′Ak′ · · · B1A1B0 be their alternating decomposition. Then α <D β ⇐ ⇒            k < k′ or, k = k′ and ∃i such that Bk = bk, Ak = ak, · · · Ai+1 = ai+1. Φ(Bi) <D Φ(bi) or , Φ(Bi) = Φ(bi) and Ai <D ai

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

• f 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

• rdering <D (of Bn−1).

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

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 length ℓ, whether 1 <D β or not is determined in time O(ℓ2).

Tetsuya Ito Braid calculus Aug , 2014 61 / 64

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 length ℓ, whether 1 <D β or not is determined in time O(ℓ2). Iteration of alternating decomposition provides nice enumeration of positive braids with respect to <D, and allows us to interpret <D as lexicographical ordering:

Corollary (Burckel ’97, I. ’10 )

The well-ordered set (B+

n , <D) is of type ωωn−2.

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 A = B+

n−1 and B = ∆(B+ n−1)∆−1.

In the dual case, we use A1 = B+∗

n−1, A2 = δB+∗ n−1δ−1, A3 = δ2B+∗ n−1δ−2, . . .

• 2. The Dehornoy ordering is a speicial one of the Nielsen-Thurston type
• rdering. 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.

• r, 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.

Tetsuya Ito Braid calculus Aug , 2014 64 / 64