Knot commensurability and the Berge conjecture

Details Knot commensurability and the Berge conjecture Knot commensurability and the Berge conjecture

2010-08-05

arxiv.org/abs/1008.1034v3

Mathematics

Geometric Topology

v3: This version is reorganized with minor errors fixed. Proposition 4.1, Corollary 4.2, and Proposition 5.8 were added. Quest

Scientific paper

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.

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