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 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.
Keywords
Cite
@article{arxiv.1008.1034,
title = {Knot commensurability and the Berge conjecture},
author = {Michel Boileau and Steven Boyer and Radu Cebanu and Genevieve S. Walsh},
journal= {arXiv preprint arXiv:1008.1034},
year = {2014}
}
Comments
v3: This version is reorganized with minor errors fixed. Proposition 4.1, Corollary 4.2, and Proposition 5.8 were added. Question 7.2 was upgraded to Theorem 7.2. 30 pages, 1 figure