Conjugacy Classes of 3-Braid Group B_3

TL;DR

This paper studies the conjugacy classes in the 3-braid group B_3, identifying summit sets, minimal elements, and computing the Hilbert series, with applications to classifying knots with braid index up to three.

Contribution

It provides a detailed description of conjugacy classes in B_3, including summit sets and Hilbert series, linking algebraic properties to knot classification.

Findings

Identified summit sets in B_3

Determined smallest elements in summit sets

Computed Hilbert series for conjugacy classes

Abstract

In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.