Knot commensurability and the Berge conjecture

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 fibred with the same genus and are chiral. A characterization of cyclic commensurability classes of complements of periodic knots is also given. In the nonperiodic case, we reduce the characterization of cyclic commensurability classes to a generalization of the Berge conjecture.