Higher degree covering moves for 3-manifolds

TL;DR

This paper extends the set of covering moves for 3-manifolds to fixed degrees d ≥ 4, showing minimal moves needed to relate colored links and classifying certain branched covers as lens spaces.

Contribution

It provides a complete set of covering moves for fixed degree d ≥ 4 and establishes minimal tangle replacements needed for link equivalence after stabilization.

Findings

Two local tangle replacements suffice after stabilization for d ≥ 4.

Any two colored links can be related by two moves after stabilization.

The d-fold simple branched cover of a d-bridge knot is a lens space L(p,q).

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 $q$.