Geometry & Topology

Knot commensurability and the Berge conjecture

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

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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.

Article information

Geom. Topol., Volume 16, Number 2 (2012), 625-664.

Received: 8 February 2011
Accepted: 27 November 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M10: Covering spaces 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

hyperbolic knot commensurability


Boileau, Michel; Boyer, Steven; Cebanu, Radu; Walsh, Genevieve S. Knot commensurability and the Berge conjecture. Geom. Topol. 16 (2012), no. 2, 625--664. doi:10.2140/gt.2012.16.625.

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