(A) fast algorithm to the conjugacy problem on generic braids

Abstract

The subjects of this thesis are the algorithmic solution to the conjugacy problem on generic braids and the solution to the reducibility problem. In chapter 1, we introduce the conjugacy problem on braid groups and three dynamic types of braids. In chapter 2, We study random braids that are formed by multiplying randomly chosen permutation braids by analyzing their behavior under 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 show that some power of a pseudo-Anosov braid is always cyclically weighted up to cycling and we also give upper bounds for the necessary exponent and the necessary number of iterated cyclings. In chapter 3, We propose an algorithm for deciding the dynamic type of a given braid. The algorithm is based on Garside`s weighted decomposition and is polynomial-time in the word-length of an input braid. Moreover a reduction system of circles can be found if the input is a certain type of reducible braids.