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Knot commensurability and the Berge conjecture

Michel Boileau, Steven Boyer, Radu Cebanu and Genevieve S Walsh

Geometry & Topology 16 (2012) 625–664

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.

hyperbolic knot, commensurability
Mathematical Subject Classification 2010
Primary: 57M10, 57M25
Received: 8 February 2011
Accepted: 27 November 2011
Published: 18 April 2012
Proposed: Cameron Gordon
Seconded: Walter Neumann, Colin Rourke
Michel Boileau
Institut de Mathématiques de Toulouse
Université Paul Sabatier
118 route de Narbonne
F-31062 Toulouse Cedex 9
Steven Boyer
Département de mathématiques
Université du Québec à Montréal
PO Box 8888
Montréal QC H3C 3P8
Radu Cebanu
Département de mathématiques
Université du Québec à Montréal
PO Box 8888
Montréal QC H3C 3P8
Genevieve S Walsh
Department of Mathematics
Tufts University
503 Boston Ave
Medford MA 02155