Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S-3 has Property I E if the infinite cyclic cover of the knot exterior embeds into S-3. Clearly all fibred knots have Property I E. There are infinitely many non-fibred knots with Property I E and infinitely many non-fibred knots without property I E. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property I E, then its Alexander polynomial Delta(k)(t) must be either 1 or 2t(2) - 5t + 2, and we give two infinite families of non-fibred genus 1 knots with Property I E and having Delta(k)(t) = 1 and 2t2 - 5t + 2 respectively. Hence among genus 1 non-fibred knots, no alternating knot has Property I E, and there is only one knot with Property I E up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold. (c) 2006 Elsevier Ltd. All rights reserved.