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Transcript of Garside structure and Dehornoy ordering of braid groups...
Garside structure and Dehornoy ordering of braidgroups for topologist (mini-course I)
Tetsuya Ito
Combinatorial Link Homology Theories, Braids, and Contact GeometryAug, 2014
Tetsuya Ito Braid calculus Sep , 2014 1 / 98
Contents
Introduction
Part I: Garside theory of braid groups
I-1: Toy model: Garside structure on Z2
I-2: Classical Garside structure
I-3: Dual Garside structure
I-4: Application to topology (1): Nielsen-Thurston classification
I-5: Application to topology (2): Curve diagram and linearrepresentation
Tetsuya Ito Braid calculus Sep , 2014 2 / 98
Introduction
Tetsuya Ito Braid calculus Sep , 2014 3 / 98
Braid group
The n-strand braid group
Bn =
⟨σ1, . . . , σn−1
σiσjσi = σiσjσi , |i − j | = 1σiσj = σjσi , |i − j | > 1
⟩.
C
t = 0
t = 1
i
1 2 i i+ 1 n1 n
An element of Bn is represented by n-strings (braid) in C× [0, 1].We have a natural map π : Bn → Sn, and the pure braid group Pn is thekernel of π.
Tetsuya Ito Braid calculus Sep , 2014 4 / 98
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
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
Braid group in topology (II) relation to MCG
Dn = z ∈ C | |z | ≤ n + 1 − 1, . . . , n: n-punctured disc
Bn∼= MCG (Dn)
= Mapping class group of Dn
= f : DnHomeo−→ Dn | f |∂Dn = id/Isotopy
σi ↔ Half Dehn-twist swapping i and (i + 1).
t = 0
t = 1
half
Dehn-twist
Tetsuya Ito Braid calculus Sep , 2014 7 / 98
Braid group in topology (II) relation to MCG
Dn = z ∈ C | |z | ≤ n + 1 − 1, . . . , n: n-punctured disc
Bn∼= MCG (Dn)
= Mapping class group of Dn
= f : DnHomeo−→ Dn | f |∂Dn = id/Isotopy
σi ↔ Half Dehn-twist swapping i and (i + 1).
t = 0
t = 1
half
Dehn-twist
Tetsuya Ito Braid calculus Sep , 2014 7 / 98
Braid group in topology (III) configuration space
The ordered/unordered configuration space of n-points in C:
Cn(C) = (z1, . . . , zn) ∈ Cn | zi = zj if i = j, UCn(C) = Cn(C)/Sn
Based loops in UCn(C) are naturally regarded as braids soΩUCn(C) = Space of braidsπ1(Cn(C)) = Pn, π1(UCn(C)) = Bn.
Tetsuya Ito Braid calculus Sep , 2014 8 / 98
Braid group in topology (III) configuration space
A natural projection
p : Cn(C)→ Cn−1(C), p(z1, . . . , zn) 7→ (z1, . . . , zn−1)
is a fibration with fiber C− (n − 1) points, with section
s : Cn−1(C)→ Cn(C), s(z1, . . . , zn−1,max|zi |+ 1).
This shows
Theorem (Atrin ’47, Fox-Neuwirth ’62, Fadell-Neuwirth ’62)
1. Cn(C) = K (Pn, 1), UCn(C) = K (Bn, 1)
2. The cohomological dimension of Bn and Pn are finite, and both Bn
and Pn are torsion-free.
3. Pn = Fn−1 ⋊ Pn−1 = (Fn−1 ⋊ (Fn−2 ⋊ (Fn−3 · · · (F2 ⋊ F1)) · · · ).
Tetsuya Ito Braid calculus Sep , 2014 9 / 98
Braid group in topology (III) configuration space
A natural projection
p : Cn(C)→ Cn−1(C), p(z1, . . . , zn) 7→ (z1, . . . , zn−1)
is a fibration with fiber C− (n − 1) points, with section
s : Cn−1(C)→ Cn(C), s(z1, . . . , zn−1,max|zi |+ 1).
This shows
Theorem (Atrin ’47, Fox-Neuwirth ’62, Fadell-Neuwirth ’62)
1. Cn(C) = K (Pn, 1), UCn(C) = K (Bn, 1)
2. The cohomological dimension of Bn and Pn are finite, and both Bn
and Pn are torsion-free.
3. Pn = Fn−1 ⋊ Pn−1 = (Fn−1 ⋊ (Fn−2 ⋊ (Fn−3 · · · (F2 ⋊ F1)) · · · ).
Tetsuya Ito Braid calculus Sep , 2014 9 / 98
Braid group in topology (IV) Hyperplane arrangement
Cn(C) is regarded as the complement of an hyperplane arrangement calledthe braid arrangement:For 1 ≤ i < j ≤ n, let
Hi ,j = Ker(zi − zj) ⊂ Cn, A = Hi ,j1≤i<j≤n
ThenCn(C) = M(A) = Cn −
∪1≤i<j≤n
Hi ,j ,
1. Reflections with respect to Hi ,j ’s forms the symmetric group Sn. Close and natural connection between root systems, Coxeter groups
and Artin groups. Source of combinatorics of braids.
2. A well-known method to construct cellular decomposition of M(A)(Salvetti complex) gives a presentation of Bn.
Tetsuya Ito Braid calculus Sep , 2014 10 / 98
Braid group in topology (IV) Hyperplane arrangement
Cn(C) is regarded as the complement of an hyperplane arrangement calledthe braid arrangement:For 1 ≤ i < j ≤ n, let
Hi ,j = Ker(zi − zj) ⊂ Cn, A = Hi ,j1≤i<j≤n
ThenCn(C) = M(A) = Cn −
∪1≤i<j≤n
Hi ,j ,
1. Reflections with respect to Hi ,j ’s forms the symmetric group Sn. Close and natural connection between root systems, Coxeter groups
and Artin groups. Source of combinatorics of braids.
2. A well-known method to construct cellular decomposition of M(A)(Salvetti complex) gives a presentation of Bn.
Tetsuya Ito Braid calculus Sep , 2014 10 / 98
Part I: Garside theory for braid groups
Tetsuya Ito Braid calculus Sep , 2014 11 / 98
Word and conjugacy problem
Word/Conjugacy Problem
For given braids α, β (as a word over σ±11 , . . . , σ±1
n−1) Determine whether α = β or not.
Determine whether α and β are conjugate or not.(and, find γ such that γβγ−1 = α.)
Since group is suited for computations (encoding), our ultimate goal is:
Algebraic link Problem
For two links (represented as closed braids),
Determine whether they are the same or not
Determine basic properties (prime/split/satellite/hyperbolic,etc...)
Word/conjugacy problem is the first step towards this problem.
Tetsuya Ito Braid calculus Sep , 2014 12 / 98
Word and conjugacy problem
Word/Conjugacy Problem
For given braids α, β (as a word over σ±11 , . . . , σ±1
n−1) Determine whether α = β or not.
Determine whether α and β are conjugate or not.(and, find γ such that γβγ−1 = α.)
Since group is suited for computations (encoding), our ultimate goal is:
Algebraic link Problem
For two links (represented as closed braids),
Determine whether they are the same or not
Determine basic properties (prime/split/satellite/hyperbolic,etc...)
Word/conjugacy problem is the first step towards this problem.
Tetsuya Ito Braid calculus Sep , 2014 12 / 98
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
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
I-1: Toy model: Garside structure on Z2
Tetsuya Ito Braid calculus Sep , 2014 14 / 98
Toy model: Garside structure on Z2
G = Z2 = ⟨x , y⟩: Free abelian group of rank twoP = xayb | a, b ≥ 0: set of “positive” elements∆ = xy = yx : Garside element
Key features:
P is a submonoid: α, β ∈ P ⇒ αβ ∈ P.
For any α ∈ G , ∆nz ∈ P for sufficiently large n.
For α = xayb, β = xa′yb
′ ∈ G , define
α ≼ β ⇐⇒ a ≤ a′ and b ≤ b′
⇐⇒ α−1β ∈ P.
Then x , y ≼ ∆.
[1,∆]Def= β ∈ G | 1 ≼ β ≼ ∆ = 1, x , y ,∆.
Tetsuya Ito Braid calculus Sep , 2014 15 / 98
Toy model: Garside structure on Z2
G = Z2 = ⟨x , y⟩: Free abelian group of rank twoP = xayb | a, b ≥ 0: set of “positive” elements∆ = xy = yx : Garside element
Key features:
P is a submonoid: α, β ∈ P ⇒ αβ ∈ P.
For any α ∈ G , ∆nz ∈ P for sufficiently large n.
For α = xayb, β = xa′yb
′ ∈ G , define
α ≼ β ⇐⇒ a ≤ a′ and b ≤ b′
⇐⇒ α−1β ∈ P.
Then x , y ≼ ∆.
[1,∆]Def= β ∈ G | 1 ≼ β ≼ ∆ = 1, x , y ,∆.
Tetsuya Ito Braid calculus Sep , 2014 15 / 98
Normal form for Z2
Definition
For β ∈ G , the normal form of β is a word over x , y ,∆±1
N(β) = ∆ps1s2 · · · sr (p ∈ Z, si ∈ x , y ,∆)
such that
1. ∆−pβ ∈ P.
2. si is the ≼-largest element among x , y ,∆ satisfying
si ≼ (s−1i−1 · · · s
−11 ∆−p)β
(So normal form of β = xayb is actually written as:
N(β) =
∆ayb−a b ≥ a
∆bxa−b a ≥ b
Tetsuya Ito Braid calculus Sep , 2014 16 / 98
Normal form for Z2
Definition
For β ∈ G , the normal form of β is a word over x , y ,∆±1
N(β) = ∆ps1s2 · · · sr (p ∈ Z, si ∈ x , y ,∆)
such that
1. ∆−pβ ∈ P.
2. si is the ≼-largest element among x , y ,∆ satisfying
si ≼ (s−1i−1 · · · s
−11 ∆−p)β
(So normal form of β = xayb is actually written as:
N(β) =
∆ayb−a b ≥ a
∆bxa−b a ≥ b
Tetsuya Ito Braid calculus Sep , 2014 16 / 98
What is the meaning of normal form ?
Idea
Normal form = path in the Cayley graph which approaches the destinationin the “fastest” way at any intermediate time.
Q: How to go back to home from university ?
x
y
University
Home
We are tired, so we want to go back to home as early as possible...
Tetsuya Ito Braid calculus Sep , 2014 17 / 98
What is the meaning of normal form ?
Idea
Normal form = path in the Cayley graph which approaches destination inthe “fastest” way at any intermediate time.
Q: How to go back to home from university ?
x
y
University
Home
yyxyyx
This path is not effective (geodesic) – we can do several short-cuts.
Tetsuya Ito Braid calculus Sep , 2014 18 / 98
What is the meaning of normal form ?
Idea
Normal form = path in the Cayley graph which approaches the destinationin the “fastest” way at any intermediate time.
Q: How to go back to home from university ?
x
y
University
Home
yy
v.s.
yy
These paths are both geodesic (so the total arrival time is the same) but...
Tetsuya Ito Braid calculus Sep , 2014 19 / 98
What is the meaning of normal form ?
Idea
Normal form = path in the Cayley graph which approaches destination inthe “fastest” way at any intermediate time.
Q: How to go back to home from university ?
x
y
University
Home
yy
v.s.
yy
Normal form
After 2minutes, normal form path lies closer than other path.
Tetsuya Ito Braid calculus Sep , 2014 20 / 98
How to computing normal forms ?
Strategy to get normal form
1. By considering ∆nβ for sufficiently large n, we assume β ∈ P .
2. Starting at the final destination, we do: let us look sub-path si si+1: check whether this sub-path is “nice” of
not (whether this sub-path is a normal form or not) If this sub-path is not nice (i.e. we are going by a roundabout route)
replace this sub-path si si+1 with better one (tighten locally).
Crucial fact
By resolving local roundabouts, we will eventually get globally nice path,the normal form.
(cf. Length of geodesic connecting two point x , y in Riemannian manifold= distance of x and y)
Tetsuya Ito Braid calculus Sep , 2014 21 / 98
How to computing normal forms ?
Strategy to get normal form
1. By considering ∆nβ for sufficiently large n, we assume β ∈ P .
2. Starting at the final destination, we do: let us look sub-path si si+1: check whether this sub-path is “nice” of
not (whether this sub-path is a normal form or not) If this sub-path is not nice (i.e. we are going by a roundabout route)
replace this sub-path si si+1 with better one (tighten locally).
Crucial fact
By resolving local roundabouts, we will eventually get globally nice path,the normal form.
(cf. Length of geodesic connecting two point x , y in Riemannian manifold= distance of x and y)
Tetsuya Ito Braid calculus Sep , 2014 21 / 98
Computing normal forms: example
x
y
xyyyx
Tetsuya Ito Braid calculus Sep , 2014 22 / 98
Computing normal forms: example
x
y
xyyyx
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 23 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 24 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 25 / 98
Computing normal forms: example
x
y
xyy
+
xyyy
Tetsuya Ito Braid calculus Sep , 2014 26 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 27 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 28 / 98
Computing normal forms: example
x
y
The best hoi e
of the rst path
k
The rst letter of
the normal form
Tetsuya Ito Braid calculus Sep , 2014 29 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 30 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 31 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 32 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 33 / 98
Computing normal forms: example
x
y
The best hoi e
of the se ond path
k
The se ond letter of
the normal form
Tetsuya Ito Braid calculus Sep , 2014 34 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 35 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 36 / 98
Computing normal forms: example
x
y
xyy
+
xyy
Tetsuya Ito Braid calculus Sep , 2014 37 / 98
Computing normal forms: example
x
y
xyy
Tetsuya Ito Braid calculus Sep , 2014 38 / 98
Computing normal forms: example
x
y
xyy
+
y
Tetsuya Ito Braid calculus Sep , 2014 39 / 98
Computing normal forms: example
x
y
The normal form is:
y
Tetsuya Ito Braid calculus Sep , 2014 40 / 98
Computing normal forms: conclusion
How fast can we compute the normal form ?Previous argument says:
Conclusion
For β ∈ G of length ℓ (as a word over x , y ,∆), after performingℓ(ℓ−1)
2 = O(ℓ2) times of “local tightening” (replacing local roundaboutroute with the best one), we are able to get a normal form of β.
Moreover, note that in the process of local tightening, we just need to lookat the path of length two. This says
Conclusion’
To compute normal form, we only need finite data (of which path isbetter).
Tetsuya Ito Braid calculus Sep , 2014 41 / 98
Computing normal forms: conclusion
How fast can we compute the normal form ?Previous argument says:
Conclusion
For β ∈ G of length ℓ (as a word over x , y ,∆), after performingℓ(ℓ−1)
2 = O(ℓ2) times of “local tightening” (replacing local roundaboutroute with the best one), we are able to get a normal form of β.
Moreover, note that in the process of local tightening, we just need to lookat the path of length two. This says
Conclusion’
To compute normal form, we only need finite data (of which path isbetter).
Tetsuya Ito Braid calculus Sep , 2014 41 / 98
I-2: Classical Garside structure
Tetsuya Ito Braid calculus Sep , 2014 42 / 98
General idea of Garside structure
We want to generalize idea and method for “toy model” for more generaland complicated group G – what we need ?In toy model, we have used:
1. Submonoid P consisting of “positive elements”:P consists of positive words over certain generating sets x , y , . . . , of G .
The notion of positive elements yields a subword ordering ≼:
α ≼ βDef⇐⇒ α−1β ∈ P.
2. Special element ∆: For any β ∈ G , ∆nβ ∈ P for sufficiently large n. x , y , . . . ≼ ∆.
By giving “good” ∆ and P , one can generalize the toy model idea.
Tetsuya Ito Braid calculus Sep , 2014 43 / 98
General idea of Garside structure
We want to generalize idea and method for “toy model” for more generaland complicated group G – what we need ?In toy model, we have used:
1. Submonoid P consisting of “positive elements”:P consists of positive words over certain generating sets x , y , . . . , of G .
The notion of positive elements yields a subword ordering ≼:
α ≼ βDef⇐⇒ α−1β ∈ P.
2. Special element ∆: For any β ∈ G , ∆nβ ∈ P for sufficiently large n. x , y , . . . ≼ ∆.
By giving “good” ∆ and P , one can generalize the toy model idea.
Tetsuya Ito Braid calculus Sep , 2014 43 / 98
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
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
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
Simple braids
≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ32︸︷︷︸
Positive braids
∆ = σ1 ∨ σ2 ∨ · · · ∨ σn−1.
definition
A simple braid is a braid that satisfies 1 ≼ x ≼ ∆.
Note: B+n = Product of σ1, . . . , σn−1
= Product of simple braids
Proposition
[1,∆]Def= simple braids 1:1←→ Sn
(so simple braids are often called premutation braids)
Tetsuya Ito Braid calculus Sep , 2014 46 / 98
Simple braids
≼ is a “subword” ordering: σ2σ3 ≼ σ2σ3 σ1σ32︸︷︷︸
Positive braids
∆ = σ1 ∨ σ2 ∨ · · · ∨ σn−1.
definition
A simple braid is a braid that satisfies 1 ≼ x ≼ ∆.
Note: B+n = Product of σ1, . . . , σn−1
= Product of simple braids
Proposition
[1,∆]Def= simple braids 1:1←→ Sn
(so simple braids are often called premutation braids)
Tetsuya Ito Braid calculus Sep , 2014 46 / 98
Example: B3 case
∆ = (σ1σ2)σ1 = σ2σ1σ2, so
[1,∆] = 1, σ1, σ2, σ1σ2, σ2σ1,∆
Simple braids: each strand positively crosses with other strands at mostonce.
Tetsuya Ito Braid calculus Sep , 2014 47 / 98
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−1i−1 · · · x
−11 ∆−pβ).
By absorbing first few ∆ terms in x1, . . ., N(β) is uniquely written as
N(β) = ∆px1x2 · · · xr (p ∈ Z, xi = ∆).
We define the infimum, supremum of β by
inf(β) = p, sup(β) = p + r .
Tetsuya Ito Braid calculus Sep , 2014 48 / 98
How to compute normal form ?
As in the toy model case, a word is a normal form if and only if it is locallya normal form:
Theorem (Elrifai-Morton, Thurston)
A wordN ′(β) = ∆px1x2 · · · xr (p ∈ Z, xi ∈ [1,∆])
is a normal form if and only if
(xixi+1) ∧∆ = xi for all i
(i.e., xixi+1 is also a normal form)
Tetsuya Ito Braid calculus Sep , 2014 49 / 98
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 thecenter of Bn, so
· · ·σ−1i · · · = · · ·∆
−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1
i )︸ ︷︷ ︸Positive braid
· · ·
2. Apply local tightening repeatedly: for i = r , . . . , 1 rewrite eachsub-path xixi+1 so that it is a normal form
xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆
Tetsuya Ito Braid calculus Sep , 2014 50 / 98
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 thecenter of Bn, so
· · ·σ−1i · · · = · · ·∆
−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1
i )︸ ︷︷ ︸Positive braid
· · ·
2. Apply local tightening repeatedly: for i = r , . . . , 1 rewrite eachsub-path xixi+1 so that it is a normal form
xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆
Tetsuya Ito Braid calculus Sep , 2014 50 / 98
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 thecenter of Bn, so
· · ·σ−1i · · · = · · ·∆
−2∆2σ−1i · · · = ∆−2 · · · (∆2σ−1
i )︸ ︷︷ ︸Positive braid
· · ·
2. Apply local tightening repeatedly: for i = r , . . . , 1 rewrite eachsub-path xixi+1 so that it is a normal form
xixi+1 = x ′i x′i+1, x ′i = (xixi+1) ∧∆
Tetsuya Ito Braid calculus Sep , 2014 50 / 98
Simple example
Let us compute the normal form of a 3-braid β = (σ−12 )(σ1σ2)(σ2)(σ1σ2).
1. Rewriting β as the form ∆p (positive braids):
β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)
2. Apply local tightenings for
β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)
to get normal forms
Tetsuya Ito Braid calculus Sep , 2014 51 / 98
Simple example
Let us compute the normal form of a 3-braid β = (σ−12 )(σ1σ2)(σ2)(σ1σ2).
1. Rewriting β as the form ∆p (positive braids):
β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)
2. Apply local tightenings for
β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)
to get normal forms
Tetsuya Ito Braid calculus Sep , 2014 51 / 98
Simple example
Let us compute the normal form of a 3-braid β = (σ−12 )(σ1σ2)(σ2)(σ1σ2).
1. Rewriting β as the form ∆p (positive braids):
β = ∆−1(σ1σ2)(σ1σ2)(σ2)(σ1σ2)
2. Apply local tightenings for
β′ = (σ1σ2)(σ1σ2)(σ2)(σ1σ2)
to get normal forms
Tetsuya Ito Braid calculus Sep , 2014 51 / 98
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
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
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
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
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
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
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
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
For es to have se ond rossings
between two strands
Tetsuya Ito Braid calculus Sep , 2014 54 / 98
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 simplebraids [1,∆] is
ℓ[1,∆](β) = maxsup(β), 0 −mininf(β), 0.
Tetsuya Ito Braid calculus Sep , 2014 55 / 98
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 simplebraids [1,∆] is
ℓ[1,∆](β) = maxsup(β), 0 −mininf(β), 0.
Tetsuya Ito Braid calculus Sep , 2014 55 / 98
Normal form produces automatic structure
The characterizing property of normal form is “local”(we only need to see consecutive factor xixi+1)
Theorem (Thurston, Charney, Dehornoy)
The normal forms of Bn provides a geodesic automatic structure. Inparticular,
Set of normal forms 1:1←→ Path of certain graph (automata)
Tetsuya Ito Braid calculus Sep , 2014 56 / 98
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
Example: Automata for B3
1
1
2
1
1
1
2
2
1
2
1
2
2
1
Normal formN(β) = ∆−1∆−1(σ2σ1)(σ1σ2)(σ2σ1)
Tetsuya Ito Braid calculus Sep , 2014 58 / 98
Conjugacy problem (I)
Using normal form technique, we can solve the conjugacy (search)problem.
Basic strategy
For given α ∈ Bn, try to determine the set of “simplest” normal formsamong its conjugacy class, called ... summit set.
S(α) =
β
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
Conjugacy problem (II)
By cycling and decycling operation, we may find simpler normal formamong the conjugacy class of given braid β:
N() =
p
x
1
x
r1
x
r
=
p
x
1
p
p
x
2
x
r
x
r
p
x
1
x
r1
p
x
2
x
r1
p
x
1
p
k
p
(
p
x
r
p
)x
1
x
r1
p
0
x
0
1
x
0
r
0
: simpler normal form
de y ling
y ling
It may happen p′ > p or r ′ < r
Tetsuya Ito Braid calculus Sep , 2014 60 / 98
Conjugacy problem (II)
Theorem (Garside, ElRifai-Morton, Gebhardt, Gonzalez-Meneses)
Let α ∈ Bn.
1. By applying cycling and decylings finitely many times, we can findone element in S(α).
2. Staring from one element β ∈ S(α), by repeatedly computing theconjugate of β by simple elements, we can find all elements of S(α):
In particular, we have an algorithm to solve the conjugacy decision andproblem (determine α ∼conj α
′) and the conjugacy search problem (find βsuch that α = β−1α′β).
Tetsuya Ito Braid calculus Sep , 2014 61 / 98
Conjugacy problem (II) example of “... (summit) set”The super summit set
SS(α) =
β
β is conjugate to α withmaximal inf,minimum sup
The ultra summit set
US(α) = β ∈ SS(α) | closed under cycling operation
USSS
Tetsuya Ito Braid calculus Sep , 2014 62 / 98
Conjugacy problem (III)
Using idea of summit set, we can solve the conjugacy problem (but in timeO(e length), in general):
Computing a normal form is easy (done in polynomial time).
Starting from α, finding one element of S(α) is (conjecturally) donein polynomial time.
Size of S(α) might be quite huge – the size of S(α) might beO(e length) (So computing whole S(α) might require exponentialtimes...)
Problem
Find polynomial time algorithm for conjugacy problem of braids.
Problem
Understand the structure of summit sets.
Tetsuya Ito Braid calculus Sep , 2014 63 / 98
Conjugacy problem (III)
Using idea of summit set, we can solve the conjugacy problem (but in timeO(e length), in general):
Computing a normal form is easy (done in polynomial time).
Starting from α, finding one element of S(α) is (conjecturally) donein polynomial time.
Size of S(α) might be quite huge – the size of S(α) might beO(e length) (So computing whole S(α) might require exponentialtimes...)
Problem
Find polynomial time algorithm for conjugacy problem of braids.
Problem
Understand the structure of summit sets.
Tetsuya Ito Braid calculus Sep , 2014 63 / 98
I-3: Dual Garside structure
Tetsuya Ito Braid calculus Sep , 2014 64 / 98
Dual Garside structureThe braid group has another Garside structure called dual Garsidestructure, by consdiering different P (the set of positive elements) and δ(Garside element)
Definition
For 1 ≤ i < j ≤ n, let
ai ,j = (σi+1 · · ·σj−2σj−1)−1σi (σi+1 · · ·σj−2σj−1)
The generating set Σ∗ = ai ,j1≤i<j≤n is called a dual Garside generator(Birman-Ko-Lee generator or band generator).
1 i j i jn
a
i;j
twisted band
Tetsuya Ito Braid calculus Sep , 2014 65 / 98
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
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
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 normalform)
N∗(β) = δpd1d2 · · · dr (p ∈ Z, xi ∈ [1, δ])
which is characterized by
1. p = minn ∈ Z | δnβ ∈ B+∗n
2. xi = δ ∧∗ (d−1i−1 · · · d
−11 δ−pβ).
We define the dual supremum, dual infimum of β by
sup ∗(β) = p + r , inf ∗(β) = p
Tetsuya Ito Braid calculus Sep , 2014 67 / 98
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 wordwith respect to the length ℓ[1,δ]. In particular,
ℓ[1,δ](β) = maxsup ∗(β), 0 −mininf ∗(β), 0.
By the similar method, one can use dual normal form to solve theconjugacy problem.
Tetsuya Ito Braid calculus Sep , 2014 68 / 98
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
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
A geometric understanding of dual simple elementsLet us identify Bn with MCG (Dn). Then,
Proposition (Bessis)
Set of convex polygons in Dn1:1←→ [1, δ]
(Convex polygons is understood as non-crossing partition of n-points)
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
A geometric understanding of the normal form conditionLike classical Garside case, we have geometric useful interpretation of thenormal form condition δ ∧∗ (xy) = x .
Proposition
For x , y ∈ [1, δ],
δ ∧∗ (xy) = x ⇐⇒ Corresponding convex polygons x are “linked” to y
yx
Linked Not Linked
xxy
y
Tetsuya Ito Braid calculus Sep , 2014 71 / 98
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 numericalcorrespondence of several data of the Garside structures (numbers ofatoms, simple elements, ...) and there is no theoretical “duality” at all !
Tetsuya Ito Braid calculus Sep , 2014 72 / 98
I-3: Application to topology (1)Nielsen-Thurston classification
Tetsuya Ito Braid calculus Sep , 2014 73 / 98
Nielsen-Thurston theory
According to the dynamics of Bn∼= MCG (Dn), a braid β viewed as a
homeomorphism, β : Dn → Dn is classified into one of the following threetypes: Periodic, reducible, pseudo-Anosov
1: Periodic
βn = ∆2m for some n,m ∈ Z(i.e., Powers of β = Dehn twists along ∂Dn)
2: Reducible
β(C ) = C for some essential simple closed curves C ⊂ Dn
(A simple curve is essential ⇐⇒ C encloses more than one punctures andis not isotopic to ∂Dn)
Tetsuya Ito Braid calculus Sep , 2014 74 / 98
Nielsen-Thurston theory
According to the dynamics of Bn∼= MCG (Dn), a braid β viewed as a
homeomorphism, β : Dn → Dn is classified into one of the following threetypes: Periodic, reducible, pseudo-Anosov
1: Periodic
βn = ∆2m for some n,m ∈ Z(i.e., Powers of β = Dehn twists along ∂Dn)
2: Reducible
β(C ) = C for some essential simple closed curves C ⊂ Dn
(A simple curve is essential ⇐⇒ C encloses more than one punctures andis not isotopic to ∂Dn)
Tetsuya Ito Braid calculus Sep , 2014 74 / 98
Nielsen-Thurston theory
According to the dynamics of Bn∼= MCG (Dn), a braid β viewed as a
homeomorphism, β : Dn → Dn is classified into one of the following threetypes: Periodic, reducible, pseudo-Anosov
1: Periodic
βn = ∆2m for some n,m ∈ Z(i.e., Powers of β = Dehn twists along ∂Dn)
2: Reducible
β(C ) = C for some essential simple closed curves C ⊂ Dn
(A simple curve is essential ⇐⇒ C encloses more than one punctures andis not isotopic to ∂Dn)
Tetsuya Ito Braid calculus Sep , 2014 74 / 98
Nielsen-Thurston theory
3: Pseudo-Anosovβ is a pseudo-Anosov homomorphism (locally, there are β is λ-expandingin one direction and λ-shrinking in transverse direction for some λ > 1(This λ is called the dilatation)
Tetsuya Ito Braid calculus Sep , 2014 75 / 98
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 asolution of this problem (Bestvina-Handel algorithm).
Now, Garside theory (normal form) provides alternative solution !
Tetsuya Ito Braid calculus Sep , 2014 76 / 98
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 asolution of this problem (Bestvina-Handel algorithm).
Now, Garside theory (normal form) provides alternative solution !
Tetsuya Ito Braid calculus Sep , 2014 76 / 98
Nielsen-Thurston type via Garside theoryRecognizing a periodic braid is easy:
Theorem (Eilenberg, Kerekjarto)
A periodic n-braid is conjugate to (σ1σ2 · · ·σn−1)m or (σ1σ2 · · ·σn−1σ1)
m.In particular, if β is periodic, then βn or β(n−1) is a power of ∆2.
The problem is how to recognize a reducible braid.
Why recognizing reducible braid is not so easy ? Because, β may preservevery,very,very complicated “simple” (so it is not simple – rather complex!!!) closed curve.
Idea
Assume β is reducible. If N(β) is simple among its conjugacy class, then βpreserves “simple” (not complicated, near “standard”) simple closedcurves.
Simple normal form⇐⇒ Preserving “simple” simple closed curve
Tetsuya Ito Braid calculus Sep , 2014 77 / 98
Nielsen-Thurston type via Garside theoryRecognizing a periodic braid is easy:
Theorem (Eilenberg, Kerekjarto)
A periodic n-braid is conjugate to (σ1σ2 · · ·σn−1)m or (σ1σ2 · · ·σn−1σ1)
m.In particular, if β is periodic, then βn or β(n−1) is a power of ∆2.
The problem is how to recognize a reducible braid.
Why recognizing reducible braid is not so easy ? Because, β may preservevery,very,very complicated “simple” (so it is not simple – rather complex!!!) closed curve.
Idea
Assume β is reducible. If N(β) is simple among its conjugacy class, then βpreserves “simple” (not complicated, near “standard”) simple closedcurves.
Simple normal form⇐⇒ Preserving “simple” simple closed curve
Tetsuya Ito Braid calculus Sep , 2014 77 / 98
Easy, but informative observation
Observation
For simple braids x , y , if xy is a normal form preserving “standard” roundcurve patterns, then x and y also preserves such a curve pattern.
Tetsuya Ito Braid calculus Sep , 2014 78 / 98
Nielsen-Thurston type via Garside theory
Theorem (Barnadete-Nitecki-Gutierrez ’95)
If β is reducible, then there exists α ∈ US(β) ⊂ SS(β) such that αpreserves standard a round curve. Thus by computing US(β) or SS(β), wecan determine whether β is reducible or not.
Proof: If β is reducible, by conjugating, β preserves standard round curve.By previous observation, (de)cycling of β has the same property.
Tetsuya Ito Braid calculus Sep , 2014 79 / 98
Nielsen-Thurston type via Garside theory
Drawback
The theorem says at least one element in US(β) is very nice (preservesround curves). But, computing all US(β) may be hard (may requireexponential time !)
Reasonably-sounding result
An element of US(β) has the “simplest” normal form, so if β is reducible,elements of all US(β) preserves the simplest, a standard round curve.
This is true under some assumptions (Lee-Lee ’08), but is not true ingeneral: (think appropriate simple element, for example)
Tetsuya Ito Braid calculus Sep , 2014 80 / 98
Nielsen-Thurston type via Garside theory
Drawback
The theorem says at least one element in US(β) is very nice (preservesround curves). But, computing all US(β) may be hard (may requireexponential time !)
Reasonably-sounding result
An element of US(β) has the “simplest” normal form, so if β is reducible,elements of all US(β) preserves the simplest, a standard round curve.
This is true under some assumptions (Lee-Lee ’08), but is not true ingeneral: (think appropriate simple element, for example)
Tetsuya Ito Braid calculus Sep , 2014 80 / 98
Fast Nielsen-Thurston type via Garside theory
Theorem (Gonzalez-Meneses, Wiest ’11)
If β is reducible, then after taking m-th power βm for some m < n6, everyelement in α ∈ SC (βm) preserves either standard round curves or, almostround curves. (Here SC ⊂ US is a sliding circuit, a more refinement of theUltra summit set)
Round Almost Round
Conclusion
Having simple normal form (simple in algebraic prospect) = Having simplereduction curve (simple in geomteric prospect),
Tetsuya Ito Braid calculus Sep , 2014 81 / 98
Fast Nielsen-Thurston type via Garside theory
Moreover, by applying linear bounded conjugator property
Theorem (Mazur-Minsky ’00, Tao ’13)
If x , y ∈ Bn are conjugate, then x = wxw−1, where the length of w ∈ Bn
is at most Constant C (n) · (length of x + y)))
We have (theoretically fast) algorithm:
Theorem (Calvez ’14)
By using Garside theory machinery, one can determine whether β isreducible or not in quadratic time.
Remark
Unfortunately, due to the lack of our knowledge of precise value of C (n),the algorithm in thw above theorem is not practical at this moment.
Tetsuya Ito Braid calculus Sep , 2014 82 / 98
Questions
At this moment, our argument recognizes periodic and reducible braids.
Problem
Can we recoginze/understand pseudo-Anosov braid (dilatation, theirinvariant train-track) from Garside theory ?
A reasonably-sounding idea is that if α is pseudo-Anosov and β ∈ SS(α),then the invariant train-track of β is simple in some sense.
Remark
For a pseudo-Anosov braid β, then there exists m < n6 such that thenormal form of βm has certain nice property called rigidity.
Tetsuya Ito Braid calculus Sep , 2014 83 / 98
I-5: Application to topology (2): Curvediagram and linear representation
Tetsuya Ito Braid calculus Sep , 2014 84 / 98
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 boundaryand the first puncture, and define
Γβ = (Γ− e)β
Tetsuya Ito Braid calculus Sep , 2014 85 / 98
Labelling of Curve diagram I: winding number labelling
Make curve diagram so that it has minimum vertical tangencies, andassign labelling (winding number labelling) as follows: if we turn clockwisedirection, add +1 and if we turn counter-clockwise direction, add −1
i
(i + 1)
i
(i + 1)
(i + 1)
i
i
(i + 1)
0
1
0
1
Tetsuya Ito Braid calculus Sep , 2014 86 / 98
Labelling of Curve diagram II: wall-crossing numberlabelling
Make curve diagram so that it has minimum intersection with walls(vertical line from punctures) and that near the puncture it is horizontal.Assign labelling wall crossing labelling by signed counting of intersectionswith walls (here we escape puncture in counter-clockwise direction).
i (i + 1)
(i + 1) i
0
1
2
2
1
0
1
Tetsuya Ito Braid calculus Sep , 2014 87 / 98
Labelling of Curve diagram and Garside theory
Theorem (Wiest ’09)
1. max Winding number labelling on Γβ = sup(β)
2. min Winding number labelling on Γβ = inf(β)
(Classical Garside normal form measures “how many times the braid βwinds real axis”)
Theorem (I-Wiest ’12)
1. max Wall crossing number labelling on Γβ = sup ∗(β)
2. min Wall crossing number labelling on Γβ = inf ∗(β)
(Dual Garside normal form measures “how many times the image of thereal axis across the walls”)
Tetsuya Ito Braid calculus Sep , 2014 88 / 98
Labelling of Curve diagram and Garside theory
Theorem (Wiest ’09)
1. max Winding number labelling on Γβ = sup(β)
2. min Winding number labelling on Γβ = inf(β)
(Classical Garside normal form measures “how many times the braid βwinds real axis”)
Theorem (I-Wiest ’12)
1. max Wall crossing number labelling on Γβ = sup ∗(β)
2. min Wall crossing number labelling on Γβ = inf ∗(β)
(Dual Garside normal form measures “how many times the image of thereal axis across the walls”)
Tetsuya Ito Braid calculus Sep , 2014 88 / 98
Sketch of proofStrategy:
Multiply inverse of (dual) simple elements so that maximum labellingdecreases
This process provides an effective (fastest) way to make the braidtrivial by using (dual) simple elements⇒ it is the meaning of normal form!
Here we give a proof for dual case: we isotope curve diagram and wall sothat it has circular symmetry (wall-corssing labelling does not change).
Tetsuya Ito Braid calculus Sep , 2014 89 / 98
Sketch of proofStrategy:
Multiply inverse of (dual) simple elements so that maximum labellingdecreases
This process provides an effective (fastest) way to make the braidtrivial by using (dual) simple elements⇒ it is the meaning of normal form!
Here we give a proof for dual case: we isotope curve diagram and wall sothat it has circular symmetry (wall-corssing labelling does not change).
Tetsuya Ito Braid calculus Sep , 2014 89 / 98
Sketch of proofStrategy:
Multiply inverse of (dual) simple elements so that maximum labellingdecreases
This process provides an effective (fastest) way to make the braidtrivial by using (dual) simple elements⇒ it is the meaning of normal form!
Here we give a proof for dual case: we isotope curve diagram and wall sothat it has circular symmetry (wall-corssing labelling does not change).
Tetsuya Ito Braid calculus Sep , 2014 89 / 98
Sketch of proof
The set of arcs in curve diagram with maximal wall-crossing labellingsuggests which dual simple element is needed to simplify the diagram: the“convex hull” of maximally labelled arcs provides the most economicaluntangling dual simple element.
Tetsuya Ito Braid calculus Sep , 2014 90 / 98
Lawrence-Krammer-Bigelow representation
C : Configration space of two points in Dn
C = (z1, z2) ∈ D2n | z1 = z2/(z1, z2) ≡ z2, z1)
then H1(C ;Z) = Zn ⊕ Z =⊕⟨xi ⟩ ⊕ ⟨t⟩, where
xi : meridian of hypersurface z1 = i-th puncturet : meridian of hypersurface z1 = z2
Let π : C → C be the Z2-cover associated with the kernel of
α : π1(C )Hurewicz→ H1(C ;Z)→ Z2 ∼= ⟨x⟩ ⊕ ⟨t⟩ (xi 7→ x , t 7→ t).
H2(C ;Z) is a free Z[x±1, t±1]-module of rank(n2
).
Tetsuya Ito Braid calculus Sep , 2014 91 / 98
Lawrence-Krammer-Bigelow representation
C : Configration space of two points in Dn
C = (z1, z2) ∈ D2n | z1 = z2/(z1, z2) ≡ z2, z1)
then H1(C ;Z) = Zn ⊕ Z =⊕⟨xi ⟩ ⊕ ⟨t⟩, where
xi : meridian of hypersurface z1 = i-th puncturet : meridian of hypersurface z1 = z2
Let π : C → C be the Z2-cover associated with the kernel of
α : π1(C )Hurewicz→ H1(C ;Z)→ Z2 ∼= ⟨x⟩ ⊕ ⟨t⟩ (xi 7→ x , t 7→ t).
H2(C ;Z) is a free Z[x±1, t±1]-module of rank(n2
).
Tetsuya Ito Braid calculus Sep , 2014 91 / 98
Lawrence-Krammer-Bigelow representation
The braid group Bn = MCG (Dn) action on Dn induces an action on C (upto homotopy), so we get
ρLKB : Bn → GL(H2(C ;Z))
called the Lawrence-Krammer-Bigelow representation. By choosingappropriate basis vi ,j1≤i<j≤n coming from topology, the LKBrepresentation is given by
ρLKB(σi )(vj ,k) =
Fj ,k i ∈ j − 1, j , k − 1, kqFi ,k + (q2 − q)Fi ,j + (1− q)Fj ,k i = j − 1
Fj+1,k i = j = k − 1
qFj ,i + (1− q)Fj ,k + (q − q2)tFi ,k i = k − 1 = j
Fj ,k+1 i = k
−q2tFj ,k i = j = k − 1
Tetsuya Ito Braid calculus Sep , 2014 92 / 98
Lawrence-Krammer-Bigelow representation
Surprisingly, Lawrence-Krammer-Bigelow representation detects the normalforms.
Theorem (Krammer ’02, I-Wiest ’12)
For β ∈ Bn,
1. maxdegree of t in the matrix ρLKB(β) = sup(β).
2. mindegree of t in the matrix ρLKB(β) = inf(β)
3. maxdegree of q in the matrix ρLKB(β) = 2 sup ∗(β).
4. mindegree of q in the matrix ρLKB(β) = 2 inf ∗(β)
Corollary (Krammer, Bigelow ’02)
The Lawrence-Krammer-Bigelow representation is faithful – so, the braidgroups are linear.
Tetsuya Ito Braid calculus Sep , 2014 93 / 98
Why LKB representation know the Garside structures ?
Compare the definition of α : π1(C )→ ⟨x⟩ ⊕ ⟨t⟩ with the definition oflabelling of curve diagram:
Labelling = Position of the lift of the curve ∼= variables q and t.
D
n
0 1
0
1
e
f
D
n
q
q
Tetsuya Ito Braid calculus Sep , 2014 94 / 98
Quantum representation
By theory of quantum group, for a Uq(g)-module V , (quantum envelopingalgebra of semi-simple lie algebra g), we have a linear representation calledquantum representations
ρV : Bn → GL(V⊗n)
that is a q-deformation of permutation
ϕV : Sn → GL(V⊗n),
(i , i + 1)(v1 ⊗ · · · ⊗ vi−1 ⊗ vi ⊗ · · · ⊗ vn) = v1 ⊗ · · · ⊗ vi+1 ⊗ vi ⊗ vn
Quantum representations are important because they produces invariantsof knto and 3-manifolds, called Quantum invariants.
Tetsuya Ito Braid calculus Sep , 2014 95 / 98
Quantum representation and invariants
Braids
ρVQuantum
representation
Closure // (Oriented) Links Surgery//
Quantuminvariant
Closed 3-manifolds
Quantuminvariant
GL(V⊗n)
“Trace′′ // C[q, q−1]q=e
2π√
−1N
Take linear sums// C
Tetsuya Ito Braid calculus Sep , 2014 96 / 98
Quantum representation and Garside theory
Using KZ-equation argument (realizing quantum representation as certainmonodromy representation), one identifies “generic” quantum sl2representation with homological representation similar toLawrence-Krammer-Bigelow representation (Kohno,I, Jackson-Kerler).Then, we have:
Theorem (I. ’12)
For β ∈ Bn. the maximal and the minimal degree of weight variable in“Generic” quantum sl2-representation is equal to the constant multiples ofsup ∗(β) and inf ∗(β).
⇒ Quantum representation (quantum group) is also related to (dual)Garside structure.
Tetsuya Ito Braid calculus Sep , 2014 97 / 98
Problems
Problem
Find a relationship between linear representations and the classical Garsidestructure: Conjecturally, it should be related to the quantum parameter q.
Problem
Find a relationship between quantum knots or3-manifold invariants (forexample, Jones polynomial) and Garside theory.
Problem
Find a direct, more conceptual understanding between quantumrepresentation and Garside structure.
Tetsuya Ito Braid calculus Sep , 2014 98 / 98