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

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

2 Contents Introduction Part I: Garside theory of braid groups I-1: Toy model: Garside structure on Z 2 I-2: Classical Garside structure I-3: Dual Garside structure I-4: Application to topology (1): Nielsen-Thurston classification I-5: Application to topology (2): Curve diagram and linear representation Tetsuya Ito Braid calculus Sep, / 98

3 Introduction Tetsuya Ito Braid calculus Sep, / 98

4 Braid group The n-strand braid group σ B n = σ 1,..., σ i σ j σ i = σ i σ j σ i, i j = 1 n 1 σ i σ j = σ j σ i, i j > 1. Ø ¼ ½ ¾ ½ Ò ½ Ò Ø ½ An element of B n is represented by n-strings (braid) in C [0, 1]. We have a natural map π : B n S n, and the pure braid group P n is the kernel of π. Tetsuya Ito Braid calculus Sep, / 98

5 Braid group in topology (I) relation to knot theory Alexander-Markov Theorem {Braids}/conjugation,stabilization) 1:1 { Oriented links in S 3 } ««½ Tetsuya Ito Braid calculus Sep, / 98

6 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 S 3 } ««½ Tetsuya Ito Braid calculus Sep, / 98

7 Braid group in topology (II) relation to MCG D n = {z C z n + 1} {1,..., n}: n-punctured disc B n = MCG(Dn ) = {Mapping class group of D n } = {f : D n Homeo D n f Dn = id}/{isotopy} Tetsuya Ito Braid calculus Sep, / 98

8 Braid group in topology (II) relation to MCG D n = {z C z n + 1} {1,..., n}: n-punctured disc B n = MCG(Dn ) = {Mapping class group of D n } = {f : D n Homeo D n f Dn = id}/{isotopy} σ i Half Dehn-twist swapping i and (i + 1). Ø ¼ Ð Ò¹ØÛ Ø Ø ½ Tetsuya Ito Braid calculus Sep, / 98

9 Braid group in topology (III) configuration space The ordered/unordered configuration space of n-points in C: C n (C) = {(z 1,..., z n ) C n z i z j if i j}, UC n (C) = C n (C)/S n Based loops in UC n (C) are naturally regarded as braids so { ΩUC n (C) = {Space of braids} π 1 (C n (C)) = P n, π 1 (UC n (C)) = B n. Tetsuya Ito Braid calculus Sep, / 98

10 Braid group in topology (III) configuration space A natural projection p : C n (C) C n 1 (C), p(z 1,..., z n ) (z 1,..., z n 1 ) is a fibration with fiber C {(n 1) points}, with section s : C n 1 (C) C n (C), s(z 1,..., z n 1, max{ z i } + 1). This shows Tetsuya Ito Braid calculus Sep, / 98

11 Braid group in topology (III) configuration space A natural projection p : C n (C) C n 1 (C), p(z 1,..., z n ) (z 1,..., z n 1 ) is a fibration with fiber C {(n 1) points}, with section This shows s : C n 1 (C) C n (C), s(z 1,..., z n 1, max{ z i } + 1). Theorem (Atrin 47, Fox-Neuwirth 62, Fadell-Neuwirth 62) 1. C n (C) = K(P n, 1), UC n (C) = K(B n, 1) 2. The cohomological dimension of B n and P n are finite, and both B n and P n are torsion-free. 3. P n = F n 1 P n 1 = (F n 1 (F n 2 (F n 3 (F 2 F 1 )) ). Tetsuya Ito Braid calculus Sep, / 98

12 Braid group in topology (IV) Hyperplane arrangement C n (C) is regarded as the complement of an hyperplane arrangement called the braid arrangement: For 1 i < j n, let H i,j = Ker(z i z j ) C n, A = {H i,j } 1 i<j n Then C n (C) = M(A) = C n H i,j, 1 i<j n Tetsuya Ito Braid calculus Sep, / 98

13 Braid group in topology (IV) Hyperplane arrangement C n (C) is regarded as the complement of an hyperplane arrangement called the braid arrangement: For 1 i < j n, let H i,j = Ker(z i z j ) C n, A = {H i,j } 1 i<j n Then C n (C) = M(A) = C n H i,j, 1 i<j n 1. Reflections with respect to H i,j s forms the symmetric group S n. 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 B n. Tetsuya Ito Braid calculus Sep, / 98

14 Part I: Garside theory for braid groups Tetsuya Ito Braid calculus Sep, / 98

15 Word and conjugacy problem Word/Conjugacy Problem For given braids α, β (as a word over {σ ±1 1,..., σ±1 n 1 }) Determine whether α = β or not. Determine whether α and β are conjugate or not. (and, find γ such that γβγ 1 = α.) Tetsuya Ito Braid calculus Sep, / 98

16 Word and conjugacy problem Word/Conjugacy Problem For given braids α, β (as a word over {σ ±1 1,..., σ±1 n 1 }) Determine whether α = β or not. Determine whether α and β are conjugate or not. (and, find γ such that γβγ 1 = α.) Since group is suited for computations (encoding), our ultimate goal is: Algebraic link Problem For two links (represented as closed braids), Determine whether they are the same or not Determine basic properties (prime/split/satellite/hyperbolic,etc...) Word/conjugacy problem is the first step towards this problem. Tetsuya Ito Braid calculus Sep, / 98

17 What is Garside theory? Garside theory (Garside structure) is a machinery of: 1. Producing the normal form of a group. Easy to calculate (and suited for computor) Idea and its meaning sounds natural. 2. Giving several nice structures of the group (automatic, lattice...) 3. Allowing us to solve other problems (conjugacy, extracting roots, etc...) In particular, for the case of braid groups: Tetsuya Ito Braid calculus Sep, / 98

18 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, / 98

19 I-1: Toy model: Garside structure on Z 2 Tetsuya Ito Braid calculus Sep, / 98

20 Toy model: Garside structure on Z 2 G = Z 2 = x, y : Free abelian group of rank two P = {x a y b a, b 0}: set of positive elements = xy = yx: Garside element Tetsuya Ito Braid calculus Sep, / 98

21 Toy model: Garside structure on Z 2 G = Z 2 = x, y : Free abelian group of rank two P = {x a y b 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, n z P for sufficiently large n. For α = x a y b, β = x a y b G, define Then x, y. α β a a and b b α 1 β P. [1, ] Def = {β G 1 β } = {1, x, y, }. Tetsuya Ito Braid calculus Sep, / 98

22 Normal form for Z 2 Definition For β G, the normal form of β is a word over {x, y, ±1 } N(β) = p s 1 s 2 s r (p Z, s i {x, y, }) such that Tetsuya Ito Braid calculus Sep, / 98

23 Normal form for Z 2 Definition For β G, the normal form of β is a word over {x, y, ±1 } N(β) = p s 1 s 2 s r (p Z, s i {x, y, }) such that 1. p β P. 2. s i is the -largest element among {x, y, } satisfying s i (s 1 i 1 s 1 1 p )β (So normal form of β = x a y b is actually written as: { a y b a b a N(β) = b x a b a b Tetsuya Ito Braid calculus Sep, / 98

24 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? ÀÓÑ Ý ÍÒ Ú Ö ØÝ Ü We are tired, so we want to go back to home as early as possible... Tetsuya Ito Braid calculus Sep, / 98

25 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? ÀÓÑ Ý ÝÝÜ ÝÝÜ ÍÒ Ú Ö ØÝ Ü This path is not effective (geodesic) we can do several short-cuts. Tetsuya Ito Braid calculus Sep, / 98

26 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? ÀÓÑ Ý ÝÝ Úº º ÝÝ ÍÒ Ú Ö ØÝ Ü These paths are both geodesic (so the total arrival time is the same) but... Tetsuya Ito Braid calculus Sep, / 98

27 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? ÀÓÑ Ý ÝÝ Úº º ÝÝ ÆÓÖÑ Ð ÓÖÑ ÍÒ Ú Ö ØÝ Ü After 2minutes, normal form path lies closer than other path. Tetsuya Ito Braid calculus Sep, / 98

28 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 si 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, / 98

29 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 si 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, / 98

30 Computing normal forms: example Ý Ü ÝÝ ÝÜ Ü Tetsuya Ito Braid calculus Sep, / 98

31 Computing normal forms: example Ý Ü Ü ÝÝ ÝÜ Ü ÝÝ Tetsuya Ito Braid calculus Sep, / 98

32 Computing normal forms: example Ý Ü ÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

33 Computing normal forms: example Ý Ü Ü ÝÝ Ü Ý Ý Tetsuya Ito Braid calculus Sep, / 98

34 Computing normal forms: example Ý Ü Ü Ý Ý Ü ÝÝÝ Tetsuya Ito Braid calculus Sep, / 98

35 Computing normal forms: example Ý Ü ÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

36 Computing normal forms: example Ý Ü Ü ÝÝ Ü ÝÝ Tetsuya Ito Braid calculus Sep, / 98

37 Computing normal forms: example Ý Ü Ì Ø Ó Ó Ø Ö Ø Ô Ø Ì Ö Ø Ð ØØ Ö Ó Ø ÒÓÖÑ Ð ÓÖÑ Tetsuya Ito Braid calculus Sep, / 98

38 Computing normal forms: example Ý Ü Ü ÝÝ Ü Ý Ý Tetsuya Ito Braid calculus Sep, / 98

39 Computing normal forms: example Ý Ü Ü Ý Ý Ü ÝÝ Tetsuya Ito Braid calculus Sep, / 98

40 Computing normal forms: example Ý Ü ÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

41 Computing normal forms: example Ý Ü Ü ÝÝ Ü ÝÝ Tetsuya Ito Braid calculus Sep, / 98

42 Computing normal forms: example Ý Ü Ì Ø Ó Ó Ø ÓÒ Ô Ø Ì ÓÒ Ð ØØ Ö Ó Ø ÒÓÖÑ Ð ÓÖÑ Tetsuya Ito Braid calculus Sep, / 98

43 Computing normal forms: example Ý Ü ÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

44 Computing normal forms: example Ý Ü ÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

45 Computing normal forms: example Ý Ü Ü ÝÝ ÜÝÝ Tetsuya Ito Braid calculus Sep, / 98

46 Computing normal forms: example Ý ÜÝÝ Ü Tetsuya Ito Braid calculus Sep, / 98

47 Computing normal forms: example Ý Ü ÜÝÝ Ý Tetsuya Ito Braid calculus Sep, / 98

48 Computing normal forms: example Ý Ì ÒÓÖÑ Ð ÓÖÑ Ý Ü Tetsuya Ito Braid calculus Sep, / 98

49 Computing normal forms: conclusion How fast can we compute the normal form? Previous argument says: Conclusion For β G of length l (as a word over {x, y, }), after performing l(l 1) 2 = O(l 2 ) times of local tightening (replacing local roundabout route with the best one), we are able to get a normal form of β. Tetsuya Ito Braid calculus Sep, / 98

50 Computing normal forms: conclusion How fast can we compute the normal form? Previous argument says: Conclusion For β G of length l (as a word over {x, y, }), after performing l(l 1) 2 = O(l 2 ) times of local tightening (replacing local roundabout route with the best one), we are able to get a normal form of β. Moreover, note that in the process of local tightening, we just need to look at the path of length two. This says Conclusion To compute normal form, we only need finite data (of which path is better). Tetsuya Ito Braid calculus Sep, / 98

51 I-2: Classical Garside structure Tetsuya Ito Braid calculus Sep, / 98

52 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, / 98

53 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 : α β Def α 1 β P. 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, / 98

54 The classical Garside structure of braid B + n = {Product of σ 1,..., σ n 1 } : Positive braid monoid = (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, / 98

55 The classical Garside structure of braid B + n = {Product of σ 1,..., σ n 1 } : Positive braid monoid = (σ 1 σ 2 σ n 1 )(σ 1 σ 2 σ n 2 ) (σ 1 σ 2 )(σ 1 ) : 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 σ 1, σ 2,..., σ n 1. x y = min {z B n x, y z} Tetsuya Ito Braid calculus Sep, / 98

56 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 and s 1 should be: p β B + n 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, / 98

57 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, / 98

58 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 + n = {Product of σ 1,..., σ n 1 } = {Product of simple braids} Proposition [1, ] Def = { simple braids } 1:1 S n (so simple braids are often called premutation braids) Tetsuya Ito Braid calculus Sep, / 98

59 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, / 98

60 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, / 98

61 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 is a normal form if and only if (i.e., x i x i+1 is also a normal form) (p Z, x i [1, ]) (x i x i+1 ) = x i for all i Tetsuya Ito Braid calculus Sep, / 98

62 How to compute normal form? The strategy for computing normal form applies to the braid group case: Strategy to get normal form 1. Express β as a word of the form β = p x 1 x r (p Z, x i [1, ]) Tetsuya Ito Braid calculus Sep, / 98

63 How to compute normal form? The strategy for computing normal form applies to the braid group case: Strategy to get normal form 1. Express β as a word of the form β = 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 σ 1 i = 2 2 σ 1 i = 2 ( 2 σ 1 i ) }{{} Positive braid Tetsuya Ito Braid calculus Sep, / 98

64 How to compute normal form? The strategy for computing normal form applies to the braid group case: Strategy to get normal form 1. Express β as a word of the form β = 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 σ 1 i = 2 2 σ 1 i = 2 ( 2 σ 1 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 i+1, x i = (x i x i+1 ) Tetsuya Ito Braid calculus Sep, / 98

65 Simple example Let us compute the normal form of a 3-braid β = (σ 1 2 )(σ 1σ 2 )(σ 2 )(σ 1 σ 2 ). Tetsuya Ito Braid calculus Sep, / 98

66 Simple example Let us compute the normal form of a 3-braid β = (σ 1 2 )(σ 1σ 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, / 98

67 Simple example Let us compute the normal form of a 3-braid β = (σ 1 2 )(σ 1σ 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 to get normal forms β = (σ 1 σ 2 )(σ 1 σ 2 )(σ 2 )(σ 1 σ 2 ) Tetsuya Ito Braid calculus Sep, / 98

68 Simple example: local tightening β = (σ 1 σ 2 )(σ 1 σ 2 )(σ 2 )(σ 1 σ 2 ) Tetsuya Ito Braid calculus Sep, / 98

69 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, / 98

70 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, / 98

71 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, / 98

72 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, / 98

73 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, / 98

74 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, / 98

75 Meaning of normal form condition The situation F (x) S(y) prevents to absorb crossings in y into x: (Recall that: simple braid each pair of strand crosses at most by once ¾ ½ µ ½ Ë Ë ½ ¾ µ ½ ÓÖ ØÓ Ú ÓÒ ÖÓ Ò ØÛ Ò ØÛÓ ØÖ Ò Tetsuya Ito Braid calculus Sep, / 98

76 Geodesic property Lemma x 1 and x = x 1 are simple if x is simple. Tetsuya Ito Braid calculus Sep, / 98

77 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, / 98

78 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, {Set of normal forms} 1:1 {Path of certain graph (automata)} Tetsuya Ito Braid calculus Sep, / 98

79 Example: Automata for B 3 ¾ ½ ½ ½ ½ ¾ ¾ ½ ½ ¾ ½ ½ ¾ ¾ Tetsuya Ito Braid calculus Sep, / 98

80 Example: Automata for B 3 ¾ ½ ½ ½ ½ ¾ ¾ ½ ½ ¾ ½ ½ ¾ ¾ Normal form N(β) = 1 1 (σ 2 σ 1 )(σ 1 σ 2 )(σ 2 σ 1 ) Tetsuya Ito Braid calculus Sep, / 98

81 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. Then, S(α) = { β β is conjugate to α, with the simplest N(β) + Additional requirements } S(α) = S(α ) α and β are conjugate Tetsuya Ito Braid calculus Sep, / 98

82 Conjugacy problem (II) By cycling and decycling operation, we may find simpler normal form among the conjugacy class of given braid β: Æ µ Ô Ü ½ Ü Ö ½ Ü Ö Ô Ü ½ Ô Ô Ü ¾ Ü Ö ÝÐ Ò ÝÐ Ò Ü Ö Ô Ü ½ Ü Ö ½ Ô Ü ¾ Ü Ö ½ Ô Ü ½ Ô Ô Ô Ü Ö Ô µü ½ Ü Ö ½ Ô¼ Ü ¼ ½ ܼ ¼ ÑÔÐ Ö ÒÓÖÑ Ð ÓÖÑ Ö It may happen p > p or r < r Tetsuya Ito Braid calculus Sep, / 98

83 Conjugacy problem (II) Theorem (Garside, ElRifai-Morton, Gebhardt, González-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, / 98

84 Conjugacy problem (II) example of... (summit) set The super summit set SS(α) = The ultra summit set { β β is conjugate to α with maximal inf, minimum sup US(α) = {β SS(α) closed under cycling operation } } SS US Tetsuya Ito Braid calculus Sep, / 98

85 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, / 98

86 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, / 98

87 I-3: Dual Garside structure Tetsuya Ito Braid calculus Sep, / 98

88 Dual Garside structure The braid group has another Garside structure called dual Garside structure, by consdiering different P (the set of positive elements) and δ (Garside element) Definition For 1 i < j n, let 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). ½ Ò ØÛ Ø Ò Tetsuya Ito Braid calculus Sep, / 98

89 Dual Garside structure B n + = {Product of positive a i,j } : Dual positive monoid 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, / 98

90 Dual Garside structure B n + = {Product of positive a i,j } : Dual positive monoid 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 a i,j δ for all 1 i < j n. x y = min {z B n x, y z} Tetsuya Ito Braid calculus Sep, / 98

91 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, / 98

92 Dual Garside structure A parallel argument applies for the dual Garside structure: Theorem (Birman-Ko-Lee) The dual normal form provides an automatic structure. Theorem (Birman-Ko-Lee) An appropriate modification of dual normal form provides a geodesic word with respect to the length l [1,δ]. In particular, l [1,δ] (β) = max{sup (β), 0} min{inf (β), 0}. By the similar method, one can use dual normal form to solve the conjugacy problem. Tetsuya Ito Braid calculus Sep, / 98

93 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, / 98

94 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, / 98

95 A geometric understanding of dual simple elements Let us identify B n with MCG(D n ). Then, Proposition (Bessis) {Set of convex polygons in D n } 1:1 [1, δ] (Convex polygons is understood as non-crossing partition of n-points) ¾ ½ ½ µ µ Tetsuya Ito Braid calculus Sep, / 98

96 A geometric understanding of the normal form condition Like classical Garside case, we have geometric useful interpretation of the normal form condition δ (xy) = x. Proposition For x, y [1, δ], δ (xy) = x Corresponding convex polygons x are linked to y x y y x x y Linked Not Linked Tetsuya Ito Braid calculus Sep, / 98

97 Open problem Open problem Are there other Garside structures (i.e. the submonoid P and element which allows us to develop a machinery for normal forms) for 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, / 98

98 I-3: Application to topology (1) Nielsen-Thurston classification Tetsuya Ito Braid calculus Sep, / 98

99 Nielsen-Thurston theory According to the dynamics of B n = MCG(Dn ), 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, / 98

100 Nielsen-Thurston theory According to the dynamics of B n = MCG(Dn ), 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 = 2m for some n, m Z (i.e., Powers of β = Dehn twists along D n ) Tetsuya Ito Braid calculus Sep, / 98

101 Nielsen-Thurston theory According to the dynamics of B n = MCG(Dn ), 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 = 2m for some n, m Z (i.e., Powers of β = Dehn twists along D n ) 2: Reducible β(c) = C for some essential simple closed curves C D n (A simple curve is essential C encloses more than one punctures and is not isotopic to D n ) Tetsuya Ito Braid calculus Sep, / 98

102 Nielsen-Thurston theory 3: Pseudo-Anosov β is a pseudo-anosov homomorphism (locally, there are β is λ-expanding in one direction and λ-shrinking in transverse direction for some λ > 1 (This λ is called the dilatation) Tetsuya Ito Braid calculus Sep, / 98

103 Nielsen-Thurston theory Knowing the Nielsen-Thurston type is important in dynamics, topology (and algebraic properties like centralizers), so Problem How to determine the Nielsen-Thurston type of β? Tetsuya Ito Braid calculus Sep, / 98

104 Nielsen-Thurston theory Knowing the Nielsen-Thurston type is important in dynamics, topology (and algebraic properties like centralizers), so Problem How to determine the Nielsen-Thurston type of β? Train-track method (graph encoding of surface automorphisms) provides a solution of this problem (Bestvina-Handel algorithm). Now, Garside theory (normal form) provides alternative solution! Tetsuya Ito Braid calculus Sep, / 98

105 Nielsen-Thurston type via Garside theory Recognizing a periodic braid is easy: Theorem (Eilenberg, Kerékjártó) A periodic n-braid is conjugate to (σ 1 σ 2 σ n 1 ) m or (σ 1 σ 2 σ n 1 σ 1 ) m. In particular, if β is periodic, then β n or β (n 1) is a power of 2. The problem is how to recognize a reducible braid. Why recognizing reducible braid is not so easy? Because, β may preserve very,very,very complicated simple (so it is not simple rather complex!!!) closed curve. Tetsuya Ito Braid calculus Sep, / 98

106 Nielsen-Thurston type via Garside theory Recognizing a periodic braid is easy: Theorem (Eilenberg, Kerékjártó) A periodic n-braid is conjugate to (σ 1 σ 2 σ n 1 ) m or (σ 1 σ 2 σ n 1 σ 1 ) m. In particular, if β is periodic, then β n or β (n 1) is a power of 2. The problem is how to recognize a reducible braid. Why recognizing reducible braid is not so easy? Because, β may preserve very,very,very complicated simple (so it is not simple rather complex!!!) closed curve. Idea Assume β is reducible. If N(β) is simple among its conjugacy class, then β preserves simple (not complicated, near standard ) simple closed curves. Simple normal form Preserving simple simple closed curve Tetsuya Ito Braid calculus Sep, / 98

107 Easy, but informative observation Observation For simple braids x, y, if xy is a normal form preserving standard round curve patterns, then x and y also preserves such a curve pattern. Tetsuya Ito Braid calculus Sep, / 98

108 Nielsen-Thurston type via Garside theory Theorem (Barnadete-Nitecki-Gutiérrez 95) If β is reducible, then there exists α US(β) SS(β) such that α preserves standard a round curve. Thus by computing US(β) or SS(β), we can determine whether β is reducible or not. Proof: If β is reducible, by conjugating, β preserves standard round curve. By previous observation, (de)cycling of β has the same property. Tetsuya Ito Braid calculus Sep, / 98

109 Nielsen-Thurston type via Garside theory Drawback The theorem says at least one element in US(β) is very nice (preserves round curves). But, computing all US(β) may be hard (may require exponential time!) Tetsuya Ito Braid calculus Sep, / 98

110 Nielsen-Thurston type via Garside theory Drawback The theorem says at least one element in US(β) is very nice (preserves round curves). But, computing all US(β) may be hard (may require exponential time!) Reasonably-sounding result An element of US(β) has the simplest normal form, so if β is reducible, elements of all US(β) preserves the simplest, a standard round curve. This is true under some assumptions (Lee-Lee 08), but is not true in general: (think appropriate simple element, for example) Tetsuya Ito Braid calculus Sep, / 98

111 Fast Nielsen-Thurston type via Garside theory Theorem (González-Meneses, Wiest 11) If β is reducible, then after taking m-th power β m for some m < n 6, every element in α SC(β m ) preserves either standard round curves or, almost round curves. (Here SC US is a sliding circuit, a more refinement of the Ultra summit set) Round Almost Round Conclusion Having simple normal form (simple in algebraic prospect) = Having simple reduction curve (simple in geomteric prospect), Tetsuya Ito Braid calculus Sep, / 98

112 Fast Nielsen-Thurston type via Garside theory Moreover, by applying linear bounded conjugator property Theorem (Mazur-Minsky 00, Tao 13) If x, y B n are conjugate, then x = wxw 1, where the length of w B n is at most Constant C(n) (length of x + y))) We have (theoretically fast) algorithm: Theorem (Calvez 14) By using Garside theory machinery, one can determine whether β is reducible or not in quadratic time. Remark Unfortunately, due to the lack of our knowledge of precise value of C(n), the algorithm in thw above theorem is not practical at this moment. Tetsuya Ito Braid calculus Sep, / 98

113 Questions At this moment, our argument recognizes periodic and reducible braids. Problem Can we recoginze/understand pseudo-anosov braid (dilatation, their invariant train-track) from Garside theory? A reasonably-sounding idea is that if α is pseudo-anosov and β SS(α), then the invariant train-track of β is simple in some sense. Remark For a pseudo-anosov braid β, then there exists m < n 6 such that the normal form of β m has certain nice property called rigidity. Tetsuya Ito Braid calculus Sep, / 98

114 I-5: Application to topology (2): Curve diagram and linear representation Tetsuya Ito Braid calculus Sep, / 98

115 Curve diagram Using identification B n = MCG(Dn ), we can represent β B n by the (isotopy class of the) image of horizontal line Γ, called Curve Diagram. µ ½ ¾ ½ ¾ (We often distinguish the first segment e of Γ connecting the boundary and the first puncture, and define Γ β = (Γ e)β Tetsuya Ito Braid calculus Sep, / 98

116 Labelling of Curve diagram I: winding number labelling Make curve diagram so that it has minimum vertical tangencies, and assign labelling (winding number labelling) as follows: if we turn clockwise direction, add +1 and if we turn counter-clockwise direction, add 1 ½µ ¼ ½ ½µ ½µ ½ ½µ ¼ Tetsuya Ito Braid calculus Sep, / 98

117 Labelling of Curve diagram II: wall-crossing number labelling Make curve diagram so that it has minimum intersection with walls (vertical line from punctures) and that near the puncture it is horizontal. Assign labelling wall crossing labelling by signed counting of intersections with walls (here we escape puncture in counter-clockwise direction). ½µ ½ ¾ ½µ ¼ ½ ¼ ¾ ½ Tetsuya Ito Braid calculus Sep, / 98

118 Labelling of Curve diagram and Garside theory Theorem (Wiest 09) 1. max {Winding number labelling on Γ β } = sup(β) 2. min {Winding number labelling on Γ β } = inf(β) (Classical Garside normal form measures how many times the braid β winds real axis ) Tetsuya Ito Braid calculus Sep, / 98

119 Labelling of Curve diagram and Garside theory Theorem (Wiest 09) 1. max {Winding number labelling on Γ β } = sup(β) 2. min {Winding number labelling on Γ β } = inf(β) (Classical Garside normal form measures how many times the braid β winds real axis ) Theorem (I-Wiest 12) 1. max {Wall crossing number labelling on Γ β } = sup (β) 2. min {Wall crossing number labelling on Γ β } = inf (β) (Dual Garside normal form measures how many times the image of the real axis across the walls ) Tetsuya Ito Braid calculus Sep, / 98

120 Sketch of proof Strategy: Multiply inverse of (dual) simple elements so that maximum labelling decreases Tetsuya Ito Braid calculus Sep, / 98

121 Sketch of proof Strategy: Multiply inverse of (dual) simple elements so that maximum labelling decreases This process provides an effective (fastest) way to make the braid trivial by using (dual) simple elements it is the meaning of normal form! Tetsuya Ito Braid calculus Sep, / 98

122 Sketch of proof Strategy: Multiply inverse of (dual) simple elements so that maximum labelling decreases This process provides an effective (fastest) way to make the braid trivial by using (dual) simple elements it is the meaning of normal form! Here we give a proof for dual case: we isotope curve diagram and wall so that it has circular symmetry (wall-corssing labelling does not change). Tetsuya Ito Braid calculus Sep, / 98

123 Sketch of proof The set of arcs in curve diagram with maximal wall-crossing labelling suggests which dual simple element is needed to simplify the diagram: the convex hull of maximally labelled arcs provides the most economical untangling dual simple element. Tetsuya Ito Braid calculus Sep, / 98

124 Lawrence-Krammer-Bigelow representation C : Configration space of two points in D n C = {(z 1, z 2 ) D 2 n z 1 z 2 }/(z 1, z 2 ) z 2, z 1 ) then H 1 (C; Z) = Z n Z = x i t, where { x i : meridian of hypersurface {z 1 = i-th puncture} t : meridian of hypersurface {z 1 = z 2 } Tetsuya Ito Braid calculus Sep, / 98

125 Lawrence-Krammer-Bigelow representation C : Configration space of two points in D n C = {(z 1, z 2 ) D 2 n z 1 z 2 }/(z 1, z 2 ) z 2, z 1 ) then H 1 (C; Z) = Z n Z = x i t, where { x i : meridian of hypersurface {z 1 = i-th puncture} t : meridian of hypersurface {z 1 = z 2 } Let π : C C be the Z 2 -cover associated with the kernel of α : π 1 (C) Hurewicz H 1 (C; Z) Z 2 = x t (xi x, t t). H 2 ( C; Z) is a free Z[x ±1, t ±1 ]-module of rank ( n 2). Tetsuya Ito Braid calculus Sep, / 98

126 Lawrence-Krammer-Bigelow representation The braid group B n = MCG(D n ) action on D n induces an action on C (up to homotopy), so we get ρ LKB : B n GL(H 2 ( C; Z)) called the Lawrence-Krammer-Bigelow representation. By choosing appropriate basis {v i,j } 1 i<j n coming from topology, the LKB representation is given by F j,k i {j 1, j, k 1, k} qf i,k + (q 2 q)f i,j + (1 q)f j,k i = j 1 F j+1,k i = j k 1 ρ LKB (σ i )(v j,k ) = qf j,i + (1 q)f j,k + (q q 2 )tf i,k i = k 1 j F j,k+1 i = k q 2 tf j,k i = j = k 1 Tetsuya Ito Braid calculus Sep, / 98

127 Lawrence-Krammer-Bigelow representation Surprisingly, Lawrence-Krammer-Bigelow representation detects the normal forms. Theorem (Krammer 02, I-Wiest 12) For β B n, 1. max{degree of t in the matrix ρ LKB (β)} = sup(β). 2. min{degree of t in the matrix ρ LKB (β)} = inf(β) 3. max{degree of q in the matrix ρ LKB (β)} = 2 sup (β). 4. min{degree of q in the matrix ρ LKB (β)} = 2 inf (β) Corollary (Krammer, Bigelow 02) The Lawrence-Krammer-Bigelow representation is faithful so, the braid groups are linear. Tetsuya Ito Braid calculus Sep, / 98

128 Why LKB representation know the Garside structures? Compare the definition of α : π 1 ( C) x t with the definition of labelling of curve diagram: Labelling = Position of the lift of the curve = variables q and t. ¼ ½ ¼ ½ Õ Ò Ò Õ Tetsuya Ito Braid calculus Sep, / 98

129 Quantum representation By theory of quantum group, for a U q (g)-module V, (quantum enveloping algebra of semi-simple lie algebra g), we have a linear representation called quantum representations ρ V : B n GL(V n ) that is a q-deformation of permutation ϕ V : S n GL(V n ), (i, i + 1)(v 1 v i 1 v i v n ) = v 1 v i+1 v i v n Quantum representations are important because they produces invariants of knto and 3-manifolds, called Quantum invariants. Tetsuya Ito Braid calculus Sep, / 98

130 Quantum representation and invariants {Braids} Closure {(Oriented) Links } Surgery {Closed 3-manifolds} ρ V GL(V n ) Quantum representation Trace C[q, q 1 ] Quantum invariant q=e 2π 1 N Take linear sums C Quantum invariant Tetsuya Ito Braid calculus Sep, / 98

131 Quantum representation and Garside theory Using KZ-equation argument (realizing quantum representation as certain monodromy representation), one identifies generic quantum sl 2 representation with homological representation similar to Lawrence-Krammer-Bigelow representation (Kohno,I, Jackson-Kerler). Then, we have: Theorem (I. 12) For β B n. the maximal and the minimal degree of weight variable in Generic quantum sl 2 -representation is equal to the constant multiples of sup (β) and inf (β). Quantum representation (quantum group) is also related to (dual) Garside structure. Tetsuya Ito Braid calculus Sep, / 98

132 Problems Problem Find a relationship between linear representations and the classical Garside structure: Conjecturally, it should be related to the quantum parameter q. Problem Find a relationship between quantum knots or3-manifold invariants (for example, Jones polynomial) and Garside theory. Problem Find a direct, more conceptual understanding between quantum representation and Garside structure. Tetsuya Ito Braid calculus Sep, / 98