Small knot complements, exceptional surgeries, and hidden symmetries

TL;DR

This paper identifies two obstructions preventing small knot complements in the 3-sphere from having hidden symmetries, advancing understanding of their symmetry properties and partially addressing a related conjecture.

Contribution

It introduces novel obstructions based on cyclic commensurability and covering properties that limit the existence of hidden symmetries in small knot complements.

Findings

Small knot complements are not cyclically commensurable with other knot complements.

Certain small knot complements cannot admit hidden symmetries if they cover manifolds with specific symmetries and multiple exceptional surgeries.

Abstract

This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture of Boileau, Boyer, Cebanu, and Walsh. We also provide a second obstruction to admitting hidden symmetries in the case where a small knot complement covers a manifold admitting some symmetry and at least two exceptional surgeries.