Michel Boileau

Institut de Mathématiques de Toulouse,
Université Paul Sabatier,
118 route de Narbonne,
31062 Toulouse Cedex 9, FRANCE.

Title: Commensurability classes of hyperbolic knot complements and hidden symmetries I.

Title: Commensurability classes of hyperbolic knot complements and hidden symmetries II.

Abstract: Two knot complements are commensurable if if they have homeomorphic finite sheeted covers. In the generic case of knots without hidden symmetries, knot complements which are commensurable are cyclically commensurable, which means that they have homeomorphic cyclic covers. Moreover there are at most three hyperbolic knot complements in a cyclic commensurability class and strong restrictions on knots with cyclic commensurable knot complements: they are fibered and chiral, with the same genus, but different volume. For the case with hidden symmetries, the situation is very different. Two knot complements with hidden symmetries may be commensurable, but not cyclically commensurable. To date, there are only three knots in $S^3$ which are known to admit hidden symmetries: the figure eight knot and the two commensurable dodecahedral knots. In these two talks we will present the results about cyclic commensurability of hyperbolic knot complements and discuss open questions and new results in the case of knots with hidden symmetries.

This is a joint work with Steve Boyer, Radu Cebanu and Genevieve Walsh.