Hurwitz equivalence in the universal dihedral quandle

TL;DR

This paper classifies the orbits of the Hurwitz action of braid groups on the universal dihedral quandle using three computable invariants, providing explicit representatives and extending results to virtual braid groups.

Contribution

Introduces three invariants that fully classify Hurwitz orbits in the universal dihedral quandle and extends the classification to virtual braid groups.

Findings

Complete classification of Hurwitz orbits using invariants

Explicit system of orbit representatives provided

Extensions to virtual braid groups achieved

Abstract

We investigate the Hurwitz action of the $m$-braid group on the $m$-fold Cartesian product of the universal dihedral quandle. We introduce three computable invariants and prove that they give a complete classification of the orbits under this action. As a consequence, we describe an explicit complete system of orbit representatives. We further obtain analogous classifications for the corresponding Hurwitz actions of the pure $m$-braid group, the virtual $m$-braid group, and the virtual pure $m$-braid group.