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Algorithmics in braid groups

Algorithmics in braid groups. Juan González-Meneses Universidad de Sevilla. Trenzas. Jornada Temática Interdisciplinar de la Red Española de Topología Barcelona , October 22 – 23, 2010. Seville. June 13-17, 2011. www.imus.us.es/ACT/braids2011. 5. 4. 3. 2. 1. Juan González-Meneses.

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Algorithmics in braid groups

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  1. Algorithmics in braid groups Juan González-Meneses Universidad de Sevilla Trenzas Jornada Temática Interdisciplinar de laRed Española de Topología Barcelona , October 22 – 23, 2010.

  2. Seville. June 13-17, 2011 www.imus.us.es/ACT/braids2011

  3. 5 4 3 2 1 Juan González-Meneses Computing in braid groups Garside structure E. Artin (1925)

  4. Juan González-Meneses Computing in braid groups Garside structure Monoid of positive braids: Induces a lattice order: Garsideelement Simple elements = Positive prefixes of 

  5. Juan González-Meneses Computing in braid groups Garside structure Garside group Dehornoy-Paris (1999) P 1 G And some otherproperties…

  6. Juan González-Meneses Computing in braid groups Garside structure Lattice of simple elements in the braid group B4.

  7. Juan González-Meneses Word problem Left normal form Artin (1925) Garside (1969), Deligne (1972), Adyan (1984), Elrifai-Morton (1988), Thurston (1992). Every braid x can be written in Left normal form: Simple elements Every product must be left-weighted. Biggestpossiblesimple elementin anysuchwriting of thisproduct.

  8. In general: Given simple elements Not left-weighted left-weighted Juan González-Meneses Word problem Left normal form a b a s t simple simple We call this procedure a local sliding applied to ab.

  9. Apply all possible local slidings,until all consecutive factors are left weighted Juan González-Meneses Word problem Left normal form Computation of a left normal form, given a product of simple elements: Left normal form: Maximal power of D. Minimal number of factors. (canonical length)

  10. Juan González-Meneses Conjugacy problem Garside (1969) ElRifai-Morton (1988) Birman-Ko-Lee (1998) Franco-GM (2003) Gebhardt (2005) Birman-Gebhardt-GM(2008) Gebhardt-GM(2009) Charney: (1992) Artin-Tits groups of spherical type are biautomatic.

  11. Juan González-Meneses Conjugacy problem The main idea Given an element x, compute the set of simplest conjugates of x. (in some sense) x and y are conjugate , their corresponding sets coincide.

  12. Juan González-Meneses Conjugacy problem Cyclic sliding Can x be simplified by a conjugation? For simplicity, we will assume that there is no power of  : Consecutive factors are left-weighted. What about x5 and x1 ? Up toconjugacy, we can considerthatx5 and x1 are consecutive

  13. Juan González-Meneses Conjugacy problem Cyclic sliding x5 x1 x4 x2 x3

  14. Juan González-Meneses Conjugacy problem Cyclic sliding s x5s x5 x5 x1 t t Cyclic sliding: x4 x4 x4 x2 x2 x2 x3 x3 x3

  15. Juan González-Meneses Conjugacy problem Cyclic sliding x5s t Theresultingbraidisnot In left normal form Cyclic sliding: x4 x2 x3 We compute itsleft normal form, and itmaybecomesimpler Iterate…

  16. sliding sliding sliding x1 … x2 x3 Juan González-Meneses Conjugacy problem Sliding circuits x Slidingcircuit of x. SC(x) Set of slidingcircuits in the Conjugacyclass of x.

  17. Juan González-Meneses Computer experiments Random braids The solution to the word problem is very efficient. The solution to the conjugacy problem seems to be very efficient. (taking braids at random) At random? Many authors do computer experiments with “random braids”.

  18. Juan González-Meneses Computer experiments Random braids Quotes from papers on braid-cryptography: …theelements ai and bi are products of 10 randomArtin generators. Garber, Kaplan, Teicher, Tsaban,Vishne. “A”chooses a randomsecretbraid… Random elements for these tests were obtained as follows. We choose independent random simple elementsA1,A2, ... untillen(A1/ Am) = r. Ko, Lee, Cheon, Han, Kang, Park. Gebhardt. …we took 5000 random pairs of positive braids in Bn of length l (using Artin presentation) Franco, González-Meneses. None of these procedures generates truly random braids!

  19. Juan González-Meneses Computer experiments Random braids We will focus on the positive braid monoid: Length of a (positive) braid = Word length We want to generate a random positive braid of length k. Whatifwetaketheproduct of krandomlychosengenerators?

  20. Juan González-Meneses Computer experiments Random braids There is only one word representing the braid There are two words representing the braid The probability of obtaining is twice the probability of obtaining There is only one word representing the braid There are 16 words representing the braid The probability of obtaining is 16 times the probability of obtaining This becomes more dramatic as n and k grow.

  21. Juan González-Meneses Generating random braids Lex-representative How do we generate braids with the same probability? Given a braida, itsLex-representative, w(a ), isthe (lexicographically) smallestwordrepresentinga. Bij. { Braids of length k } { Lex-representatives of length k }

  22. Juan González-Meneses Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B4+ : Root. The only braid of length 0 in B4+.

  23. Juan González-Meneses Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B4+ : 3 braids of length 1 in B4+.

  24. Juan González-Meneses Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B4+ : 8 braids of length 2 in B4+. The word is not there, as the Lex-representativeof is .

  25. Juan González-Meneses Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B4+ : 19 braids of length 3 in B4+. There is no subword , neither , nor .

  26. Juan González-Meneses Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B4+ : 19 braids of length 3 in B4+. There is no subword , neither , nor .

  27. Juan González-Meneses Generating random braids Generation procedure To generate a random braid: 19 1) Compute the number of leaves of the tree: 16 2) Choose a random number between 1 and 19: 3) Find the braid corresponding to the 16th leaf: In polynomial time! Warning: thetreeisexponentiallybig!

  28. Juan González-Meneses Counting braids of given length Bronfman method xn,k = Number of braids in Bn+ of length k. (want to compute this number) Recursive method to compute the growth function of Bn+ Bronfman, 2001: Wewil use anothermethod: simpler and faster.

  29. Juan González-Meneses Counting braids of given length Our method Example, in Bn+ :   = # Braids: # Braids: Cannot start with Cannot start with +   # Braids: # Braids:

  30. Juan González-Meneses Counting braids of given length Our method This yields an easy recursive formula: - + - + + =

  31. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 1 1 1 1 3 2 1 3 + - + + =

  32. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 1 1 1 1 3 2 1 3 5 2 8 8 + - + + =

  33. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 1 1 1 1 3 2 1 3 5 2 8 8 + 4 19 19 11 - + + =

  34. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 1 1 1 1 3 2 1 3 5 2 8 8 + 4 19 19 11 43 43 24 8 - + + =

  35. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 1 1 1 1 3 2 1 3 5 2 8 8 + 4 19 19 11 43 43 24 8 94 94 51 16 - + + =

  36. Juan González-Meneses Counting braids of given length Our method Example, in B4+ : 3 2 1 5 2 8 4 19 11 1 1 1 1 3 2 1 3 5 2 8 8 + 4 19 19 11 43 43 24 8 94 94 51 16 - 202 33 202 108 … + What are wecomputing? + =

  37. Juan González-Meneses Counting braids of given length Our method 3 2 1 5 2 8 4 19 11 Thefirstcolumn of ourtablecontainsx n,1 , x n,2 , x n,3 ,… Computing krows, weobtainx n,k.

  38. Juan González-Meneses Generating random braids Generation procedure To generate a random braid: 19 1) Compute the number of leaves of the tree: 16 2) Choose a random number between 1 and 19: 3) Find the braid corresponding to the 16th leaf: Next

  39. Juan González-Meneses Finding the rth braid of length k Hanging leaves Consider the graph of height k as before. Given a vertexw, supposewe can compute, in polynomial time, thenumber of leaveshangingfromw:

  40. Juan González-Meneses Finding the rth braid of length k The procedure We want to compute the 16th braid: Thefirstletteris . Thesecondletteris . Thethirdletteris .

  41. Juan González-Meneses Computing pending leaves Forbidden prefixes Computing at most (n-2)k times thehangingleaves of a vertex,one computes therthbraid of lengthk. How to compute the hanging leaves? { Leaves hanging from w } = { Lex-representatives starting with w } Lemma: (GM, 2010) Thenumber of leaveshangingfromwisequaltothenumber of braids, of lengthk-|w|, whichcannotstartwithsomeforbiddenprefixes.

  42. Juan González-Meneses Computing pending leaves Forbidden prefixes Example: Forbidden prefixes for the word Denote Know how to compute

  43. Juan González-Meneses Computing pending leaves Forbidden prefixes By the inclusion-exclusion principle, we can compute

  44. Juan González-Meneses Computing pending leaves Forbidden prefixes By the inclusion-exclusion principle, we can compute

  45. Juan González-Meneses Computing pending leaves Forbidden prefixes Good news: Every summand is equal to §xn,t, for some t·k. These are the elements in the first column of our table. Bad news:There are exponentially many summands. But we can treat the same way all elements of the form a1Ça2ÇÇas such that: ² Have the same length. ² Coincide in other technical questions, regarding permutation of strands.

  46. Juan González-Meneses Generating random positive braids The result As a consequence: One can compute in ploynomial space and time with respect to n and k. ) Theorem: (GM, 2010) Thereis a proceduretogenerate a random positive braidin Bn+ of lengthk, whose time and spacecomplexityis a polynomial in n and k.

  47. Juan González-Meneses Generating random braids From positive braids to braids Every braid can be written, in a unique way, as: where  is positive, and cannot start with . There are xn,k positive braids of length k, 1 1 1 1 3 2 1 3 xn,k-|| of them can start with . 5 2 8 8 4 19 19 11 43 43 24 8 It is easy to show that 94 94 51 16 202 33 202 108 Hence:

  48. Juan González-Meneses Generating random braids The result Theorem: (GM, 2010) Thereis a partialalgorithmtogenerate a randombraid in Bn, where | | = k, whose time and spacecomplexityis a polynomial in n and k, and whoserate of failureis < 2-n(n -1)/2. Procedure: Choose a random . Compute its left normal form. If  can start with , discardit and start again. Probability: <

  49. Juan González-Meneses Software to compute with braid groups (C++ source code) ² www.personal.us.es/meneses ² www.indiana.edu/~knotinfo/ (web applet) (CHEVIE package) ² GAP3 ² MAGMA

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