## Geometry & Topology

### 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 $3$ 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

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

Dates
Accepted: 27 November 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732405

Digital Object Identifier
doi:10.2140/gt.2012.16.625

Mathematical Reviews number (MathSciNet)
MR2928979

Zentralblatt MATH identifier
1258.57001

Keywords
hyperbolic knot commensurability

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732405

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