Conjugacy classes of periodic braids

TL;DR

This paper presents a polynomial-time algorithm for solving the conjugacy search problem in periodic braids within Garside groups, overcoming the challenge of large ultra summit sets by using partial cycling.

Contribution

It introduces a novel polynomial-time algorithm for conjugacy in periodic braids under the BKL Garside structure, despite large ultra summit sets.

Findings

Polynomial-time solution for CSP in periodic braids

Ultra summit sets can be exponentially large

Partial cycling connects conjugate braids efficiently

Abstract

Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such as the super summit set and the ultra summit set. Their complexities are proportional to the size of the finite set, even when there exist very short conjugating elements. However, ultra summit sets are very large in some cases especially for reducible braids and periodic braids. Some possible approaches to resolve this difficulty would be either to use different Garside structures and Garside groups in order to get a sufficiently small ultra summit set, or to develop an algorithm for finding a conjugating element faster than exhaustive search. Using the former method, Birman, Gonz\'alez-Meneses and Gebhardt have proposed a polynomial-time algorithm for the CSP for periodic braids. In this paper we study the conjugacy classes of periodic braids under the BKL Garside structure, and show that we can solve the CSP for periodic braids in polynomial time although their ultra summit sets are exponentially large. Our algorithm describes how to connect two periodic braids in the (possibly exponentially large) ultra summit set by applying partial cycling polynomially many times.