On the Hurwitz action on quasipositive factorizations of 3-braids

TL;DR

This paper studies the Hurwitz action on quasipositive factorizations of 3-braids, establishing orbit structure properties and providing an algorithm for finding representatives, with implications for braid classification.

Contribution

It proves that each Hurwitz orbit contains a special form element, introduces an algorithm for orbit representatives, and characterizes orbit counts for various braid classes.

Findings

Every orbit contains an element of a special form.

3-braids have finitely many Hurwitz orbits.

Positive 3-braids have a single orbit.

Abstract

We consider the Hurwitz action on quasipositive factorizations of 3-braids. We prove that every orbit contains an element of a special form. This fact provides an algorithm of finding representatives of every orbit for a given braid. We prove also that (1) any 3-braid has a finite number of orbits; (2) a Birman-Ko-Lee positive 3-braid has a single orbit; (3) a 3-braid of algebraic length two has at most two orbits.