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.

Figures (10)

• Figure 2.1: The Hurwitz action of $B_{m}$ on $\mathbb{Z}^{m}$
• Figure 4.1: Proof of Lemma \ref{['lem41']}
• Figure 4.2: The braid $\sigma_1\sigma_2^2\sigma_1$
• Figure 4.3: Three braids $\beta$, $\beta'$, and $\beta"$
• Figure 4.4: The braid $\sigma_{2k-3}\sigma_{2k-2}^2\sigma_{2k-3}$

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof