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 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.

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

References in corpus (8)

• Knot Floer homology detects fibred knots
• Holomorphic disks and link invariants
• Floer homology and surface decompositions
• Holomorphic discs and sutured manifolds
• Link Floer homology detects the Thurston norm
• Commensurability classes of 2-bridge knot complements
• Commensurability classes of (-2,3,n) pretzel knot complements
• Commensurability classes containing three knot complements

Cited by in corpus (8)

• Floer Simple Manifolds and L-Space Intervals
• Profinite rigidity and surface bundles over the circle
• Distance one lens space fillings and band surgery on the trefoil knot
• (1,1) L-space knots
• Small knot complements, exceptional surgeries, and hidden symmetries
• The Poincare homology sphere, lens space surgeries, and some knots with tunnel number two
• Non-fibered L-space knots
• Symmetries and hidden symmetries of $(\epsilon ,d_L)$-twisted knot complements