What is the meaning of normal form ? Idea Normal form = path in the Cayley graph which approaches destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ? Home y � y y x � y y x x Univ ersit y 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 destination in the “fastest” way at any intermediate time. Q: How to go back to home from university ? Home y � y y ��� v.s. ��� y y x Univ ersit y 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 in the “fastest” way at any intermediate time. Q: How to go back to home from university ? Home y � y y ��� v.s. ��� y y Normal form x Univ ersit y 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 ∆ n β for sufficiently large n , we assume β ∈ P . 2. Starting at the final destination, we do: ▶ let us look sub-path s i s i +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 s i s i +1 with better one (tighten locally). Tetsuya Ito Braid calculus Sep , 2014 21 / 98
How to computing normal forms ? Strategy to get normal form 1. By considering ∆ n β for sufficiently large n , we assume β ∈ P . 2. Starting at the final destination, we do: ▶ let us look sub-path s i s i +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 s i s i +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 y � x � y y � y x x Tetsuya Ito Braid calculus Sep , 2014 22 / 98
Computing normal forms: example y � x � y y � y x + x � y y �� x Tetsuya Ito Braid calculus Sep , 2014 23 / 98
Computing normal forms: example y � x � y y �� x Tetsuya Ito Braid calculus Sep , 2014 24 / 98
Computing normal forms: example y � x � y y � � + x � y � y � x Tetsuya Ito Braid calculus Sep , 2014 25 / 98
Computing normal forms: example y � x � y � y � + x �� y y y � x Tetsuya Ito Braid calculus Sep , 2014 26 / 98
Computing normal forms: example y � x �� y y � x Tetsuya Ito Braid calculus Sep , 2014 27 / 98
Computing normal forms: example y � x �� y y � + � x � y y � x Tetsuya Ito Braid calculus Sep , 2014 28 / 98
Computing normal forms: example y � The b est hoi e of the �rst path k The �rst letter of the no rmal fo rm x Tetsuya Ito Braid calculus Sep , 2014 29 / 98
Computing normal forms: example y � � x � y y � + � x � y � y x Tetsuya Ito Braid calculus Sep , 2014 30 / 98
Computing normal forms: example y � � x � y � y + � x �� y y x Tetsuya Ito Braid calculus Sep , 2014 31 / 98
Computing normal forms: example y � � x �� y y x Tetsuya Ito Braid calculus Sep , 2014 32 / 98
Computing normal forms: example y � �� x � y y + � x � � y y x Tetsuya Ito Braid calculus Sep , 2014 33 / 98
Computing normal forms: example y � The b est hoi e of the se ond path k The se ond letter of the no rmal fo rm x Tetsuya Ito Braid calculus Sep , 2014 34 / 98
Computing normal forms: example y � �� x � y y x Tetsuya Ito Braid calculus Sep , 2014 35 / 98
Computing normal forms: example y � �� x � y y x Tetsuya Ito Braid calculus Sep , 2014 36 / 98
Computing normal forms: example y � �� x � y y + ��� x y y x Tetsuya Ito Braid calculus Sep , 2014 37 / 98
Computing normal forms: example y � ��� xy y x Tetsuya Ito Braid calculus Sep , 2014 38 / 98
Computing normal forms: example y � ��� xy y + ���� y x Tetsuya Ito Braid calculus Sep , 2014 39 / 98
Computing normal forms: example y � The normal form is: ���� y x 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 performing ℓ ( ℓ − 1) = O ( ℓ 2 ) times of “local tightening” (replacing local roundabout 2 route with the best one), we are able to get a normal form of β . 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 performing ℓ ( ℓ − 1) = O ( ℓ 2 ) times of “local tightening” (replacing local roundabout 2 route with the best one), we are able to get a normal form of β . Moreover, note that in the process of local tightening, we just need to look at the path of length two. This says Conclusion’ To compute normal form, we only need finite data (of which path is better). Tetsuya Ito Braid calculus Sep , 2014 41 / 98
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 general and complicated group G – what we need ? In toy model, we have used: 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 general and complicated group G – what we need ? In toy model, we have used: 1. Submonoid P consisting of “positive elements”: P consists of positive words over certain generating sets { x , y , . . . , } of G . ▶ The notion of positive elements yields a subword ordering ≼ : Def ⇒ α − 1 β ∈ P . ⇐ α ≼ β 2. Special element ∆: ▶ For any β ∈ G , ∆ n β ∈ 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 + = { Product of σ 1 , . . . , σ n − 1 } : Positive braid monoid n ( σ 1 σ 2 · · · σ n − 1 )( σ 1 σ 2 · · · σ n − 2 ) · · · ( σ 1 σ 2 )( σ 1 ) : Garside element ∆ = Tetsuya Ito Braid calculus Sep , 2014 44 / 98
The classical Garside structure of braid B + = { Product of σ 1 , . . . , σ n − 1 } : Positive braid monoid n ( σ 1 σ 2 · · · σ n − 1 )( σ 1 σ 2 · · · σ n − 2 ) · · · ( σ 1 σ 2 )( σ 1 ) : Garside element ∆ = Definition-Proposition ⇒ x − 1 y ∈ B + Define the relation ≼ of B n by x ≼ y ⇐ n . Then ≼ is a lattice ordering: ▶ ≼ admits the greatest common divisor x ∧ y = max ≼ { z ∈ B n | z ≼ x , y } ▶ ≼ admits the least common multiple x ∨ y = min ≼ { z ∈ B n | x , y ≼ z } ▶ σ 1 , σ 2 , . . . , σ n − 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 ) = ∆ p s 1 · · · s r as we have done in the case Z 2 (toy model): So we first need ∆ − p β ∈ B + n and s 1 should be: the ≼ -maximal element satisfying s 1 ≼ ∆ − p β ( ∈ 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: σ 2 σ 3 ≼ σ 2 σ 3 σ 1 σ 3 2 ���� Positive braids ▶ ∆ = σ 1 ∨ σ 2 ∨ · · · ∨ σ n − 1 . Tetsuya Ito Braid calculus Sep , 2014 46 / 98
Simple braids ▶ ≼ is a “subword” ordering: σ 2 σ 3 ≼ σ 2 σ 3 σ 1 σ 3 2 ���� Positive braids ▶ ∆ = σ 1 ∨ σ 2 ∨ · · · ∨ σ n − 1 . definition A simple braid is a braid that satisfies 1 ≼ x ≼ ∆. Note: B + = { Product of σ 1 , . . . , σ n − 1 } n = { Product of simple braids } Proposition [1 , ∆] Def 1:1 = { simple braids } ← → S n (so simple braids are often called premutation braids) Tetsuya Ito Braid calculus Sep , 2014 46 / 98
Example: B 3 case ∆ = ( σ 1 σ 2 ) σ 1 = σ 2 σ 1 σ 2 , so [1 , ∆] = { 1 , σ 1 , σ 2 , σ 1 σ 2 , σ 2 σ 1 , ∆ } Simple braids: each strand positively crosses with other strands at most once. Tetsuya Ito Braid calculus Sep , 2014 47 / 98
Normal form Theorem-Definition (Garside, Elrifai-Morton, Thurston) A braid β ∈ B n admits the normal form N ( β ) = ∆ p x 1 x 2 · · · x r ( p ∈ Z , x i ∈ [1 , ∆]) where 1. ∆ − p β ∈ B + n . 2. x i = ∆ ∧ ( x − 1 i − 1 · · · x − 1 1 ∆ − p β ). By absorbing first few ∆ terms in x 1 , . . . , N ( β ) is uniquely written as N ( β ) = ∆ p x 1 x 2 · · · x r ( p ∈ Z , x i ̸ = ∆) . 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 locally a normal form: Theorem (Elrifai-Morton, Thurston) A word N ′ ( β ) = ∆ p x 1 x 2 · · · x r ( p ∈ Z , x i ∈ [1 , ∆]) is a normal form if and only if ( x i x i +1 ) ∧ ∆ = x i for all i (i.e., x i x i +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 = ∆ p x 1 · · · x r ( p ∈ Z , x i ∈ [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 = ∆ p x 1 · · · x r ( p ∈ Z , x i ∈ [1 , ∆]) ▶ ∆ 2 = ( σ 1 σ 2 · · · σ n − 1 ) n is the full-twist braid (as an element of MCG ( D n ), it is the Dehn twist along ∂ D n ), which is a generator of the center of B n , so · · · = ∆ − 2 · · · (∆ 2 σ − 1 · · · σ − 1 · · · = · · · ∆ − 2 ∆ 2 σ − 1 ) · · · i i i � �� � Positive braid 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 = ∆ p x 1 · · · x r ( p ∈ Z , x i ∈ [1 , ∆]) ▶ ∆ 2 = ( σ 1 σ 2 · · · σ n − 1 ) n is the full-twist braid (as an element of MCG ( D n ), it is the Dehn twist along ∂ D n ), which is a generator of the center of B n , so · · · = ∆ − 2 · · · (∆ 2 σ − 1 · · · σ − 1 · · · = · · · ∆ − 2 ∆ 2 σ − 1 ) · · · i i i � �� � Positive braid 2. Apply local tightening repeatedly: for i = r , . . . , 1 rewrite each sub-path x i x i +1 so that it is a normal form x i x i +1 = x ′ i x ′ x ′ i +1 , i = ( x i x i +1 ) ∧ ∆ Tetsuya Ito Braid calculus Sep , 2014 50 / 98
Simple example Let us compute the normal form of a 3-braid β = ( σ − 1 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ). Tetsuya Ito Braid calculus Sep , 2014 51 / 98
Simple example Let us compute the normal form of a 3-braid β = ( σ − 1 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ). 1. Rewriting β as the form ∆ p (positive braids): β = ∆ − 1 ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) Tetsuya Ito Braid calculus Sep , 2014 51 / 98
Simple example Let us compute the normal form of a 3-braid β = ( σ − 1 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ). 1. Rewriting β as the form ∆ p (positive braids): β = ∆ − 1 ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) 2. Apply local tightenings for β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) to get normal forms Tetsuya Ito Braid calculus Sep , 2014 51 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) ( σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, so β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )(∆) . Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) ( σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, so β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )(∆) . ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, and ( σ 1 σ 2 )(∆) = (∆)( σ 2 σ 1 ), so β ′ = ( σ 1 σ 2 )(∆)( σ 2 σ 1 ) Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) ( σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, so β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )(∆) . ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, and ( σ 1 σ 2 )(∆) = (∆)( σ 2 σ 1 ), so β ′ = ( σ 1 σ 2 )(∆)( σ 2 σ 1 ) ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, so β ′ = ∆( σ 2 σ 1 )( σ 2 σ 1 ) Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) ( σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, so β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )(∆) . ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, and ( σ 1 σ 2 )(∆) = (∆)( σ 2 σ 1 ), so β ′ = ( σ 1 σ 2 )(∆)( σ 2 σ 1 ) ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, so β ′ = ∆( σ 2 σ 1 )( σ 2 σ 1 ) ( σ 1 σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, and ( σ 1 σ 2 )( σ 1 σ 2 ) = (∆)( σ 2 ), so β ′ = ∆∆ σ 2 . Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Simple example: local tightening β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )( σ 2 )( σ 1 σ 2 ) ( σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, so β ′ = ( σ 1 σ 2 )( σ 1 σ 2 )(∆) . ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, and ( σ 1 σ 2 )(∆) = (∆)( σ 2 σ 1 ), so β ′ = ( σ 1 σ 2 )(∆)( σ 2 σ 1 ) ( σ 1 σ 2 )(∆) ∧ ∆ = ∆, so β ′ = ∆( σ 2 σ 1 )( σ 2 σ 1 ) ( σ 1 σ 2 )( σ 1 σ 2 ) ∧ ∆ = ∆, and ( σ 1 σ 2 )( σ 1 σ 2 ) = (∆)( σ 2 ), so β ′ = ∆∆ σ 2 . Hence β = ∆ − 1 β ′ = ∆ − 1 ∆∆ σ 2 and its normal form is N ( β ) = (∆)( σ 2 ) Tetsuya Ito Braid calculus Sep , 2014 52 / 98
Meaning of normal form condition What is the meaning of condition ( x i x i +1 ) ∧ ∆ = x i ? Proposition For x ∈ [1 , ∆], define the starting set S ( x ) by S ( x ) = { σ i | x = σ i · (positive braid) ( i . e . σ i ≼ x ) } and the finishing set F ( x ) by F ( x ) = { σ i | x = (positive braid) · σ i } 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 ( � � � ) = f � ; � g 2 1 3 1 3 S S ( � � � ) = f � g 1 2 3 1 F o r es to have se ond rossings b et w een t w o strands Tetsuya Ito Braid calculus Sep , 2014 54 / 98
Geodesic property Lemma x − 1 ∆ and x ∆ = ∆ x ∆ − 1 are simple if x is simple. 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 ) = ∆ p x 1 · · · x r as ∆ p x 1 · · · x r ( p > 0) (∆ − 1 x 1 ) ∆ p +1 (∆ − 1 x 2 ) ∆ p +2 · · · (∆ − 1 x − p ) x − p +1 · · · x r W ( β ) = ( p < 0 , p + r > 0) (∆ − 1 x 1 ) ∆ p +1 · · · (∆ − 1 x r ) ∆ p + r ∆ − p − r ( p + r < 0) Theorem (Charney) W ( β ) is a geodesic word. So the length of β (with respect to simple braids [1 , ∆] is ℓ [1 , ∆] ( β ) = max { sup( β ) , 0 } − min { inf( β ) , 0 } . Tetsuya Ito Braid calculus Sep , 2014 55 / 98
Normal form produces automatic structure The characterizing property of normal form is “local” (we only need to see consecutive factor x i x i +1 ) Theorem (Thurston, Charney, Dehornoy) The normal forms of B n provides a geodesic automatic structure. In particular, 1:1 { Set of normal forms } ← → { Path of certain graph (automata) } Tetsuya Ito Braid calculus Sep , 2014 56 / 98
Example: Automata for B 3 � � 1 � � 2 1 � 1 � � � � � 1 2 2 1 � 1 � 2 � � � 1 2 � 1 � 2 � Tetsuya Ito Braid calculus Sep , 2014 57 / 98
Example: Automata for B 3 � � 1 � � 2 1 � 1 � � � � � 1 2 2 1 � 1 � 2 � � � 1 2 � 1 � 2 � Normal form N ( β ) = ∆ − 1 ∆ − 1 ( σ 2 σ 1 )( σ 1 σ 2 )( σ 2 σ 1 ) 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 ∈ B n , try to determine the set of “simplest” normal forms among its conjugacy class, called ... summit set. { } β is conjugate to α, with the “simplest” N ( β ) S ( α ) = 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 form among the conjugacy class of given braid β : p p � p p N ( � ) = � x � � � x x = � x � � x � � � x 1 r � 1 r 1 2 r de y ling y ling p p p � p x � x � � � x r 1 r � 1 � x � � � x � x � 2 r � 1 1 k p � p p � (� x � ) x � � � x r 1 r � 1 0 p 0 0 � x � � � x : simpler normal form 0 1 r It may happen p ′ > p or r ′ < r Tetsuya Ito Braid calculus Sep , 2014 60 / 98
Conjugacy problem (II) Theorem (Garside, ElRifai-Morton, Gebhardt, Gonz´ alez-Meneses) Let α ∈ Bn . 1. By applying cycling and decylings finitely many times, we can find one element in S ( α ). 2. Staring from one element β ∈ S ( α ), by repeatedly computing the conjugate of β by simple elements, we can find all elements of S ( α ): In particular, we have an algorithm to solve the conjugacy decision and problem (determine α ∼ conj α ′ ) and the conjugacy search problem (find β such that α = β − 1 α ′ β ). Tetsuya Ito Braid calculus Sep , 2014 61 / 98
Conjugacy problem (II) example of “... (summit) set” The super summit set { } β is conjugate to α with SS ( α ) = β maximal inf , minimum sup The ultra summit set US ( α ) = { β ∈ SS ( α ) | closed under cycling operation } SS US Tetsuya Ito Braid calculus Sep , 2014 62 / 98
Conjugacy problem (III) Using idea of summit set, we can solve the conjugacy problem (but in time O ( 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) done in polynomial time. ▶ Size of S ( α ) might be quite huge – the size of S ( α ) might be O ( e length ) (So computing whole S ( α ) might require exponential times...) Tetsuya Ito Braid calculus Sep , 2014 63 / 98
Conjugacy problem (III) Using idea of summit set, we can solve the conjugacy problem (but in time O ( 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) done in polynomial time. ▶ Size of S ( α ) might be quite huge – the size of S ( α ) might be O ( e length ) (So computing whole S ( α ) might require exponential times...) Problem Find polynomial time algorithm for conjugacy problem of braids. Problem Understand the structure of summit sets. Tetsuya Ito Braid calculus Sep , 2014 63 / 98
I-3: Dual Garside structure Tetsuya Ito Braid calculus Sep , 2014 64 / 98
Dual Garside structure The braid group has another Garside structure called dual Garside structure, by consdiering different P (the set of positive elements) and δ (Garside element) Definition For 1 ≤ i < j ≤ n , let a i , j = ( σ i +1 · · · σ j − 2 σ j − 1 ) − 1 σ i ( σ i +1 · · · σ j − 2 σ j − 1 ) The generating set Σ ∗ = { a i , j } 1 ≤ i < j ≤ n is called a dual Garside generator (Birman-Ko-Lee generator or band generator). 1 i j n i j a t wisted band i;j Tetsuya Ito Braid calculus Sep , 2014 65 / 98
Dual Garside structure B + ∗ = { Product of positive a i , j } : Dual positive monoid n δ = σ 1 σ 2 · · · σ n − 1 = a 1 , 2 a 2 , 3 · · · a n − 1 , n : Dual Garside element Tetsuya Ito Braid calculus Sep , 2014 66 / 98
Dual Garside structure B + ∗ = { Product of positive a i , j } : Dual positive monoid n δ = σ 1 σ 2 · · · σ n − 1 = a 1 , 2 a 2 , 3 · · · a n − 1 , n : Dual Garside element Definition-Proposition Define the relation ≼ ∗ of B n by x ≼ ∗ y ⇐ ⇒ x − 1 y ∈ B + ∗ n . Then ≼ is a lattice ordering: ▶ ≼ admits the greatest common divisor x ∧ ∗ y = max ≼ ∗ { z ∈ B n | z ≼ ∗ x , y } ▶ ≼ admits the least common multiple x ∨ ∗ y = min ≼ ∗ { z ∈ B n | x , y ≼ ∗ z } ▶ a i , 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 ∈ B n | 1 ≼ ∗ β ≼ ∗ δ } = { Dual simple braids } Theorem-Definition (Birman-Ko-Lee) A braid β ∈ B n admits the unique the normal form ( dual Garside normal form) N ∗ ( β ) = δ p d 1 d 2 · · · d r ( p ∈ Z , x i ∈ [1 , δ ]) which is characterized by 1. p = min { n ∈ Z | δ n β ∈ B + ∗ n } 2. x i = δ ∧ ∗ ( d − 1 i − 1 · · · d − 1 1 δ − p β ). 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 word with respect to the length ℓ [1 ,δ ] . In particular, ℓ [1 ,δ ] ( β ) = max { sup ∗ ( β ) , 0 } − min { inf ∗ ( β ) , 0 } . By the similar method, one can use dual normal form to solve the conjugacy problem. Tetsuya Ito Braid calculus Sep , 2014 68 / 98
Dual Garside structure Example: 3-braid case δ = a 1 , 2 a 2 , 3 = a 2 , 3 a 1 , 3 = a 1 , 3 a 1 , 2 , so [1 , δ ] = { 1 , a 1 , 2 , a 2 , 3 , a 1 , 3 , δ } Recall that: Classical simple elements [1 , ∆] 1:1 ↔ Permutations S n Tetsuya Ito Braid calculus Sep , 2014 69 / 98
Dual Garside structure Example: 3-braid case δ = a 1 , 2 a 2 , 3 = a 2 , 3 a 1 , 3 = a 1 , 3 a 1 , 2 , so [1 , δ ] = { 1 , a 1 , 2 , a 2 , 3 , a 1 , 3 , δ } Recall that: Classical simple elements [1 , ∆] 1:1 ↔ Permutations S n What is the (combinatorial) meaning of dual simple elements ? To treat dual Garside elements, it is convenient to n -punctured disc D n with circular symmetry: Tetsuya Ito Braid calculus Sep , 2014 69 / 98
A geometric understanding of dual simple elements Let us identify B n with MCG ( D n ). Then, Proposition (Bessis) 1:1 { Set of convex polygons in D n } ← → [1 , δ ] (Convex polygons is understood as non-crossing partition of n -points) 3 2 4 1 5 8 6 7 ( a )( a a a ) 1 ; 3 4 ; 5 5 ; 7 7 ; 8 Tetsuya Ito Braid calculus Sep , 2014 70 / 98
A geometric understanding of the normal form condition Like classical Garside case, we have geometric useful interpretation of the normal form condition δ ∧ ∗ ( xy ) = x . Proposition For x , y ∈ [1 , δ ], δ ∧ ∗ ( xy ) = x ⇐ ⇒ Corresponding convex polygons x are “linked” to y x y x y x y Linked Not Linked 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 B n ? Open problem Clarify the meaning of the word “dual”: Currently, we use the name “dual” Garside structure because of numerical correspondence of several data of the Garside structures (numbers of atoms, simple elements, ...) and there is no theoretical “duality” at all ! Tetsuya Ito Braid calculus Sep , 2014 72 / 98
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 B n ∼ = MCG ( D n ), a braid β viewed as a homeomorphism, β : D n → D n is classified into one of the following three types: Periodic, reducible, pseudo-Anosov Tetsuya Ito Braid calculus Sep , 2014 74 / 98
Nielsen-Thurston theory According to the dynamics of B n ∼ = MCG ( D n ), a braid β viewed as a homeomorphism, β : D n → D n is classified into one of the following three types: Periodic, reducible, pseudo-Anosov 1: Periodic β n = ∆ 2 m for some n , m ∈ Z (i.e., Powers of β = Dehn twists along ∂ D n ) Tetsuya Ito Braid calculus Sep , 2014 74 / 98
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