A fast algorithm to the conjugacy problem on generic braids

Ki Hyoung Ko, Jang Won Lee

Published 2006-11-15, updated 2006-12-05Version 3

Random braids that are formed by multiplying randomly chosen permutation braids are studied by analyzing their behavior under Garside's weighted decomposition and cycling. Using this analysis, we propose a polynomial-time algorithm to the conjugacy problem that is successful for random braids in overwhelming probability. As either the braid index or the number of permutation-braid factors increases, the success probability converges to 1 and so, contrary to the common belief, the distribution of hard instances for the conjugacy problem is getting sparser. We also prove a conjecture by Birman and Gonz\'{a}lez-Meneses that any pseudo-Anosov braid can be made to have a special weighted decomposition after taking power and cycling. Moreover we give polynomial upper bounds for the power and the number of iterated cyclings required.