Criterion of Hurwitz equivalence for quasipositive factorizations of 3-braids

TL;DR

This paper develops an algorithm to determine whether two quasipositive factorizations of a 3-braid are equivalent under Hurwitz action, advancing understanding of braid factorizations and their classifications.

Contribution

It introduces a new algorithm to decide Hurwitz equivalence of quasipositive factorizations of 3-braids, building on previous finite set descriptions.

Findings

Algorithm effectively determines Hurwitz orbit membership.

Finite set contains representatives of all orbits for 3-braids.

Enhances classification methods for braid factorizations.

Abstract

We consider the Hurwitz action on quasipositive factorizations of a 3-braid. In a previous paper, for any given 3-braid we described a certain finite set which contains at least one representative of each orbit. Here we give an algorithm to decide if two elements of this finite set belong to the same orbit.