Hurwitz equivalence in the universal dihedral quandle

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.

22 pages. The original manuscript has been split into two separate papers. This version (v2) contains the first part; the second part will appear as a separate submission