Hurwitz equivalence in the universal dihedral quandle
Takuji Nakamura, Yasutaka Nakanishi, Shin Satoh, Kodai Wada · arXiv (Cornell University) · 2024
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.