Knot commensurability and the Berge conjecture

Boileau, Michel¹ ; Boyer, Steven² ; Cebanu, Radu² ; Walsh, Genevieve S³

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France
² Département de mathématiques, Université du Québec à Montréal, PO Box 8888, Centre-ville, Montréal QC H3C 3P8, Canada
³ Department of Mathematics, Tufts University, 503 Boston Ave, Medford MA 02155, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 625-664.

Résumé

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.

Détail

DOI : 10.2140/gt.2012.16.625

Classification : 57M10, 57M25
Keywords: hyperbolic knot, commensurability

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