Algorithmic recognition of quasipositive braids of algebraic length two

TL;DR

This paper presents an algorithm to determine whether a braid can be expressed as a product of two conjugates of powers of standard generators, utilizing Garside theory and cyclic sliding techniques.

Contribution

It introduces a novel algorithm for recognizing quasipositive braids of algebraic length two within braid and Garside groups, extending previous methods.

Findings

Algorithm successfully identifies such braids.

Applicable to Garside groups including Artin-Tits groups.

Leverages cyclic sliding and Garside theory for efficient recognition.

Abstract

We give an algorithm to decide if a given braid is a product of two factors which are conjugates of given powers of standard generators of the braid group. The same problem is solved in a certain class of Garside groups including Artin-Tits groups of spherical type. The solution is based on the Garside theory and, especially, on the theory of cyclic sliding developed by Gebhardt and Gonzalez-Meneses. We show that if a braid is of the required form, then any cycling orbit in its sliding circuit set in the dual Garside structure contains an element for which this fact is immediately seen from the left normal form.