Knot commensurability and the Berge conjecture

TL;DR

This paper explores the relationships between hyperbolic knot complements, showing they are cyclically commensurable under certain conditions, and connects these findings to the Berge conjecture and properties like fiberedness and chirality.

Contribution

It establishes that hyperbolic knot complements without hidden symmetries are cyclically commensurable only in limited cases and links cyclic commensurability to the Berge conjecture for non-periodic knots.

Findings

At most 3 hyperbolic knot complements in a cyclic commensurability class

Cyclically commensurable knots are fibered with the same genus and are chiral

Characterization of cyclic commensurability classes for periodic knots

Abstract

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibered with the same genus and are chiral. A characterisation of cyclic commensurability classes of complements of periodic knots is also given. In the non-periodic case, we reduce the characterisation of cyclic commensurability classes to a generalization of the Berge conjecture.