Higher degree covering moves for 3-manifolds

Abstract

Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of $S^3$ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree $d \geq 4$, extending prior work of Apostolakis and Piergallini. As a consequence we show that after stabilization to the same degree at least 4, only two local tangle replacements are required to relate any two colored links, recovering Bobtcheva and Piergallini's resolution of a conjecture of Montesinos. We also obtain that in the braided setting, the two local tangle replacements suffice after $d-2$ stabilizations. Lastly, we prove that the $d$-fold simple branched cover of a $d$-bridge knot is a lens space $L(p,q)$ and provide a method for determining $p$ and $q$.