The many faces of cyclic branched coverings of 2-bridge knots and links

TL;DR

This paper explores various descriptions and properties of cyclic branched coverings of 2-bridge knots and links, including their fundamental groups and homology, revealing their compositional structure.

Contribution

It provides multiple descriptions and presentations of these 3-manifolds, including fundamental groups and homology, and establishes their compositional structure.

Findings

Fundamental groups are cyclic in the case of 2-bridge knots.

Homology groups are computed for many cases.

Each singly-cyclic branched covering decomposes into simpler coverings.

Abstract

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, which are cyclic presentations in the case of 2-bridge knots. The homology groups are given for a wide class of cases. Moreover, we prove that each singly-cyclic branched covering of a 2-bridge link is the composition of a meridian-cyclic branched covering of a determined link and a cyclic branched covering of a trivial knot.