Post on 31-Mar-2015
Algorithmics in braid groups
Juan González-Meneses
Universidad de Sevilla
Jornada Temática Interdisciplinar de laRed Española de Topología
Barcelona , October 22 – 23, 2010.
Trenzas
Seville. June 13-17, 2011
www.imus.us.es/ACT/braids2011
Computing in braid groups Garside structure
Juan González-Meneses
1
2345
E. Artin (1925)
Computing in braid groups Garside structure
Juan González-Meneses
Monoid of positive braids:
Induces a lattice order:
Garside elementGarside element
Simple elements = Positive prefixes of
1
G
P
Computing in braid groups Garside structure
Juan González-Meneses
Garside group
Dehornoy-Paris (1999)
And some otherproperties…
Lattice of simple elements in the braid group B4.
Juan González-Meneses
Computing in braid groups Garside structure
Word problem Left normal form
Juan González-Meneses
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.
Biggest possible simple element in any such writing of this product.Biggest possible simple element in any such writing of this product.
Artin (1925)
Word problem Left normal form
Juan González-Meneses
a s t
In general: Given simple elements
a b
simple simple
Not left-weighted
left-weighted
We call this procedure a local sliding applied to a b .
Word problem Left normal form
Juan González-Meneses
Computation of a left normal form, given a product of simple elements:
Apply all possible local slidings,until all consecutive factors are left weighted
Left normal form:
Maximal power of Minimal number of factors.
(canonical length)
Conjugacy problem
Juan González-Meneses
Charney: (1992) Artin-Tits groups of spherical type are biautomatic.
Garside (1969)
ElRifai-Morton (1988)
Birman-Ko-Lee (1998)
Franco-GM (2003)
Gebhardt (2005)
Birman-Gebhardt-GM (2008)
Gebhardt-GM (2009)
Conjugacy problem The main idea
Juan González-Meneses
Given an element x, compute the set of simplest conjugates of x.
(in some sense)
x and y are conjugate , their corresponding sets coincide. x and y are conjugate , their corresponding sets coincide.
Conjugacy problem Cyclic sliding
Juan González-Meneses
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 to conjugacy, we can consider that x5 and x1 are consecutiveUp to conjugacy, we can consider that x5 and x1 are consecutive
Conjugacy problem Cyclic sliding
Juan González-Meneses
x2
x1 x 5
x4 x3
x2
x1 x 5
x4 x3
t x2
x 5
x4 x3
Conjugacy problem Cyclic sliding
Juan González-Meneses
s Cyclic sliding:
x2
x 5 s
x4 x3
t
Conjugacy problem Cyclic sliding
Juan González-Meneses
x2
x 5 s
x4 x3
t Cyclic sliding: The resulting braid is notIn left normal form
The resulting braid is notIn left normal form
We compute its left normal form, and it may become simplerWe compute its left normal form, and it may become simpler
Iterate…
Conjugacy problem Sliding circuits
Juan González-Meneses
xsliding x1
sliding x2 x3
sliding …
Sliding circuit of x.Sliding circuit of x.
SC(x) Set of sliding circuits in the
Conjugacy class of x.
Set of sliding circuits in the
Conjugacy class of x.
Computer experiments Random braids
Juan González-Meneses
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?At random?
Many authors do computer experiments with “random braids”.
Computer experiments Random braids
Quotes from papers on braid-cryptography:
Juan González-Meneses
“A” chooses a random secret braid…“A” chooses a random secret braid…
Ko, Lee, Cheon, Han, Kang, Park.
…we took 5000 random pairs of positive braids in Bn of length l (using Artin presentation)
…we took 5000 random pairs of positive braids in Bn of length l (using Artin presentation)
Franco, González-Meneses.
…the elements ai and bi are products of 10 random Artin generators.…the elements ai and bi are products of 10 random Artin generators.
Garber, Kaplan, Teicher, Tsaban,Vishne.
Random elements for these tests were obtained as follows. We choose independent random simple elements A1,A2, ... until len(A1/ Am) = r.
Random elements for these tests were obtained as follows. We choose independent random simple elements A1,A2, ... until len(A1/ Am) = r.
Gebhardt.
None of these procedures generates truly random braids!
Computer experiments Random braids
We will focus on the positive braid monoid:
Juan González-Meneses
Length of a (positive) braid = Word lengthLength of a (positive) braid = Word length
We want to generate a random positive braid of length k.
What if we take the product of k randomly chosen generators? What if we take the product of k randomly chosen generators?
Computer experiments Random braids
Juan González-Meneses
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
This becomes more dramatic as n and k grow.
There are 16 words representing the braid
There is only one word representing the braid
The probability of obtaining is 16 times the probability of obtaining
Generating random braids Lex-representative
Juan González-Meneses
How do we generate braids with the same probability?
Given a braid , its Lex-representative, w, is the (lexicographically) smallest word representing .
Given a braid , its Lex-representative, w, is the (lexicographically) smallest word representing .
{ Braids of length k } { Lex-representatives of length k }Bij.
Generating random braids Braid tree
Juan González-Meneses
Example, in B4+ :
The situation can be described using a rooted tree.
Root.
The only braid of length 0 in B4+.
Generating random braids Braid tree
Juan González-Meneses
Example, in B4+ :
The situation can be described using a rooted tree.
3 braids of length 1 in B4+.
Generating random braids Braid tree
Juan González-Meneses
Example, in B4+ :
The situation can be described using a rooted tree.
8 braids of length 2 in B4+. The word is not there,
as the Lex-representativeof is .
Generating random braids Braid tree
Juan González-Meneses
Example, in B4+ :
The situation can be described using a rooted tree.
19 braids of length 3 in B4+. There is no subword ,
neither , nor .
Generating random braids Braid tree
Juan González-Meneses
Example, in B4+ :
The situation can be described using a rooted tree.
19 braids of length 3 in B4+. There is no subword ,
neither , nor .
Generating random braids Generation procedure
Juan González-Meneses
To generate a random braid:
1) Compute the number of leaves of the tree: 19
2) Choose a random number between 1 and 19: 16
3) Find the braid corresponding to the 16th leaf: In polynomial time!In polynomial time!
Warning: the tree is exponentially big!Warning: the tree is exponentially big!
Counting braids of given length Bronfman method
Juan González-Meneses
x n,k = Number of braids in Bn+ of length k.
Recursive method to compute the growth function of Bn+
(want to compute this number)
Bronfman, 2001:
We wil use another method: simpler and faster.We wil use another method: simpler and faster.
Counting braids of given length Our method
Juan González-Meneses
Example, in Bn+ :
= # Braids: Cannot start with
# Braids: Cannot start with
+ # Braids: # Braids:
Counting braids of given length Our method
Juan González-Meneses
This yields an easy recursive formula:
+
+
+
-
-
=
Counting braids of given length Our method
Juan González-Meneses
+
+
+
-
=
Example, in B4+ :
1 1 1 1
1233
Counting braids of given length Our method
Juan González-Meneses
+
+
+
-
=
Example, in B4+ :
1 1 1 1
1233
2588
Counting braids of given length Our method
Juan González-Meneses
+
+
+
-
=
Example, in B4+ :
1 1 1 1
1233
2588
4111919
Counting braids of given length Our method
Juan González-Meneses
Example, in B4+ :
1 1 1 1
1233
2588
4111919
8244343+
+
+
-
=
Counting braids of given length Our method
Juan González-Meneses
Example, in B4+ :
1 1 1 1
1233
2588
4111919
8244343
16519494+
+
+
-
=
Counting braids of given length Our method
Juan González-Meneses
Example, in B4+ :
1 1 1 1
1233
2588
4111919
8244343
16519494
33108202202 …
What are we computing?What are we computing?
+
+
+
-
=
123
258
41119
Counting braids of given length Our method
Juan González-Meneses
123
258
41119
The first column of our table contains x n,1 , x n,2 , x n,3 ,…
Computing k rows, we obtain x n,k.Computing k rows, we obtain x n,k.
Generating random braids Generation procedure
Juan González-Meneses
To generate a random braid:
1) Compute the number of leaves of the tree: 19
2) Choose a random number between 1 and 19: 16
3) Find the braid corresponding to the 16th leaf: NextNext
Finding the rth braid of length k Hanging leaves
Juan González-Meneses
Given a vertex w, suppose we can compute, in polynomial time,
the number of leaves hanging from w:
Given a vertex w, suppose we can compute, in polynomial time,
the number of leaves hanging from w:
Consider the graph of height k as before.
Finding the rth braid of length k The procedure
Juan González-Meneses
We want to compute the 16th braid:
The first letter is .The first letter is .
The second letter is .The second letter is .
The third letter is .The third letter is .
Computing pending leaves Forbidden prefixes
Juan González-Meneses
Computing at most (n-2)k times the hanging leaves of a vertex,
one computes the rth braid of length k.
Computing at most (n-2)k times the hanging leaves of a vertex,
one computes the rth braid of length k.
How to compute the hanging leaves?
{ Leaves hanging from w } = { Lex-representatives starting with w }
Lemma: (GM, 2010) The number of leaves hanging from w is equal to
the number of braids, of length k-|w|, which cannot start with someforbidden prefixes.
Lemma: (GM, 2010) The number of leaves hanging from w is equal to
the number of braids, of length k-|w|, which cannot start with someforbidden prefixes.
Computing pending leaves Forbidden prefixes
Juan González-Meneses
Example:
Forbidden prefixes for the word
Denote
Know how to compute
Computing pending leaves Forbidden prefixes
Juan González-Meneses
By the inclusion-exclusion principle, we can compute
Computing pending leaves Forbidden prefixes
Juan González-Meneses
By the inclusion-exclusion principle, we can compute
Computing pending leaves Forbidden prefixes
Juan González-Meneses
Good news: Every summand is equal to § xn,t, for some t· k.
These are the elements in the first column of our table.
But we can treat the same way all elements of the form a1Ç a2Ç Ç as
such that: ² Have the same length.
² Coincide in other technical questions, regarding permutation of strands.
Bad news: There are exponentially many summands.
Generating random positive braids The result
Juan González-Meneses
As a consequence:
Theorem: (GM, 2010) There is a procedure to generate a random positive braid
in Bn+ of length k, whose time and space complexity is a polynomial in n and k.
Theorem: (GM, 2010) There is a procedure to generate a random positive braid
in Bn+ of length k, whose time and space complexity is a polynomial in n and k.
One can compute in ploynomial space and time with respect to n and k.
)
Every braid can be written, in a unique way, as:
Generating random braids From positive braids to braids
Juan González-Meneses
where is positive, and cannot start with .
There are xn,k positive braids of length k,
xn,k-|| of them can start with .1 1 1 1
1233
2588
4111919
8244343
16519494
33108202202Hence:
It is easy to show that
Generating random braids The result
Juan González-Meneses
Theorem: (GM, 2010) There is a partial algorithm to generate a random braid
in Bn, where | | = k, whose time and space complexity is a polynomial in n and k,
and whose rate of failure is < 2-n(n -1)/2.
Theorem: (GM, 2010) There is a partial algorithm to generate a random braid
in Bn, where | | = k, whose time and space complexity is a polynomial in n and k,
and whose rate of failure is < 2-n(n -1)/2.
Procedure: Choose a random .
Compute its left normal form.
If can start with , discard it and start again.
Probability: <
Software to compute with braid groups
Juan González-Meneses
² www.personal.us.es/meneses
² www.indiana.edu/~knotinfo/
² GAP3
² MAGMA
(C++ source code)
(web applet)
(CHEVIE package)