Embedding infinite cyclic covers of knot spaces into 3-space

Abstract

We say a knot k in the 3-sphere S^3 has Property IE if the infinite cyclic cover of the knot exterior embeds into S^3. Clearly all fibred knots have Property IE. There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δk(t) must be either 1 or 2t^2−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having _(Δk)(t)=1 and 2t^2−5t+2 respectively. Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.