Knot complements, hidden symmetries and reflection orbifolds

TL;DR

This paper investigates the Neumann-Reid conjecture, confirming that only the figure-eight and dodecahedral knots admit hidden symmetries by analyzing their covers of hyperbolic reflection orbifolds.

Contribution

It verifies the conjecture for knots covering small hyperbolic reflection orbifolds and characterizes when AP knots cover reflection orbifolds.

Findings

Only figure-eight and dodecahedral knots admit hidden symmetries.

Knots covering small hyperbolic reflection orbifolds are characterized.

AP knots cover reflection orbifolds only if they are specific known knots.

Abstract

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when $K$ is an AP knot, the complement of $K$ covers a reflection orbifold exactly when $K$ is either the figure-eight knot or one of the dodecahedral knots.