Conjugacy classes of periodic braids
Eon-Kyung Lee, Sang Jin Lee

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…
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Taxonomy
TopicsGeometric and Algebraic Topology · semigroups and automata theory · Finite Group Theory Research
