Any Knot Complement Covers at Most One Knot Complement

Given a 3-manifold M , there are generically infinitely many manifold which covers M . However, if we are restricted to the category of knot complements, the situation is quite different. It can be shown (see Lemma 1 and bellow) that if the complement E(K) of a knot K is n-fold covered by some knot complement, then the covering is cyclic, and K admits a cyclic surgery, i.e. a Dehn surgery such that the fundamental group of the resulting manifold is a cyclic group Zn. It follows from the Cyclic Surgery Theorem of [CGLS] that if K is not a torus knot, then there are at most two such coverings. The situation is also clear if K is a torus knot: By a theorem of Moser [M], a Dehn surgery on a (p, q) torus knot T (p, q) is a cyclic surgery if and only if the surgery coefficient is (kpq ± 1)/k for some k. Now the kpq± 1 fold cyclic covering of the complement E(K) of K is always homeomorphic to E(K) itself, with possibly an orientation reversing homeomorphism. So E(K) is only covered by one knot complement, although there are infinitely many different covering maps. In this paper we will study a closely related problem: How many knot complements are nontrivially covered by a given knot complement E(K)? The problem was studied by Gonzales-Acuna and Whitten in [GW], where they proved that a knot complement covers at most finitely many knot complements up to homeomorphism. The main result of this paper is