On Braid Group: More Sharpness for Lower Bound of the Number of Conjugacy Classes in Positive Permutation Braids

Abstract

This chapter identifies the lower and upper bounds for the number of conjugacy classes in positive permutation braids. Every possible valve for crossing number, such that, minimum and maximum crossing numbers, and a more sharpened lower bound on the number of their conjugacy classes in \(\mathit{S}^+_{n}\) are provided for such braids with associated type cycle (n). Also, for positive permutation braids with associate type cycle (\(\mathit{n}_1\), \(\mathit{n}_2\), ..., \(\mathit{n}_k\)) the minimum crossing number is given.