Finite group actions and cyclic branched covers of knots in $\mathbf{S}^3$

TL;DR

This paper establishes upper bounds on the number of knots in the 3-sphere that can produce a given hyperbolic 3-manifold as a cyclic branched cover, based on finite group actions on 3-manifolds.

Contribution

It proves a maximum of fifteen knots for hyperbolic 3-manifolds and six for irreducible 3-manifolds as cyclic branched covers, linking group actions to knot coverings.

Findings

Hyperbolic 3-manifolds can be cyclic branched covers of at most fifteen knots.

Irreducible 3-manifolds can be cyclic branched covers of at most six knots for odd prime orders.

Finite groups of diffeomorphisms constrain the number of such knot coverings.

Abstract

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-manifolds: an irreducible $3$-manifold can be the cyclic branched cover of odd prime order of at most six knots in $\mathbf{S}^3$.