Cusp types of quotients of hyperbolic knot complements

TL;DR

This paper classifies the possible cusp shapes in quotients of hyperbolic knot complements, showing certain types cannot occur and establishing relationships between different cusp types in orbifold covers.

Contribution

It completes the classification of cusp types in hyperbolic knot complement quotients and reveals new restrictions and relationships among cusp shapes.

Findings

$S^2(2,4,4)$ cannot be a cusp cross-section of any hyperbolic knot orbifold.

If a knot complement covers an orbifold with a $S^2(2,3,6)$ cusp, it also covers one with a $S^2(3,3,3)$ cusp.

All cusp types can appear in quotients of link complements.

Abstract

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a $S^2(2,3,6)$ cusp, it also covers an orbifold with a $S^2(3,3,3)$ cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.