The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if anybody knows of any software that can tell me if any two given braids (given their braid word) are conjugate.
$\begingroup$
$\endgroup$
2
-
2$\begingroup$ Did you see this? arxiv.org/pdf/math/0112310.pdf . The algorithm seems to be easy to implement, although I do not know if anybody has done that, $\endgroup$– user6976Commented Jan 20, 2020 at 1:53
-
2$\begingroup$ Magma claims to have this. magma.maths.usyd.edu.au/magma/handbook/text/866#10067 "Note that testing elements for conjugacy is a hard problem and may require significant amounts of memory and CPU time." magma.maths.usyd.edu.au/magma/handbook/text/869#10152 $\endgroup$– MyNinthAccountCommented Jan 20, 2020 at 2:18
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
The program flipper can solve the conjugacy problem for pseudo-Anosov elements of mapping class groups. In principle this allows one to solve the conjugacy problem in the braid group $B_n$ for many pairs of words. Suppose one has two braids whose monodromy is pseudo-Anosov in the $n$-punctured plane mapping class group. Flipper can solve the conjugacy problem in the mapping class group of the $n+1$-punctured sphere for this pair of braids, and then one must determine if the conjugacy preserves the point at infinity (this actually requires one to compute the centralizer of the mapping class group element, which is possible using the veering triangulation data flipper produces). Since the braid group is a central extension of a subgroup of the mapping class group of the $n$-punctured plane fixing infinity, two braids whose mapping class representatives are conjugate will be conjugate in the braid group if they have the same writhe.
$\begingroup$
$\endgroup$
Braidlab, a matlab package: https://arxiv.org/abs/1410.0849
(see p.19 of the PDF manual linked)
