A lower bound for the number of conjugacy classes in positive permutation braids

Abstract

We study the problem of finding lower and upper bounds for the number of conjugacy clases in positive permutation braids. For such braids with associated type cycle (n), all possible values of their crossing numbers, the minimum and maximum crossing numbers and a more sharbend lower bound of the number of their conjugacy classes in S_{n}⁺ are given. Also for positive permutation braids with associate type cycle (n₁,n₂,…,n_{k}) the minimum crossing number is given.