Conjugacy classes of positive $3$-braids

TL;DR

This paper provides a direct, explicit characterization of conjugacy classes of positive 3-braids, offering a concrete and closed-form description for all conjugates within these classes.

Contribution

It introduces a novel, explicit structural description of conjugacy classes of positive 3-braids, complementing existing algorithmic approaches.

Findings

Explicit conjugacy class descriptions for positive 3-braids

Closed-form formulas for all conjugates of a given positive 3-braid

Enhanced understanding of the structure of positive braid conjugacy classes

Abstract

The conjugacy problem in braid groups has been extensively studied, particularly from an algorithmic perspective. Established methods based on Garside structures, such as initial summit sets and super summit sets, provide effective procedures for determining whether two braids are conjugate. In contrast, explicit structural descriptions of conjugacy classes are less frequently addressed. Although cyclic sliding offers a powerful mechanism for navigating distinguished subsets within a conjugacy class, it is well known that conjugate braids cannot, in general, be obtained from one another solely through iterated cyclic sliding. In this paper, we provide a direct and explicit characterization of the conjugacy classes of positive $3$-braids. Specifically, for any given positive $3$-braid, we determine all of its conjugates in a concrete and closed form.