Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot kk in the 3-sphere S3S3 has Property  IEIE if the infinite cyclic cover of the knot exterior embeds into S3S3. Clearly all fibred knots have Property IEIE.There are infinitely many non-fibred knots with Property IEIE and infinitely many non-fibred knots without property IEIE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IEIE, then its Alexander polynomial Δk(t)Δk(t) must be either 1 or 2t2−5t+22t2−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IEIE and having Δk(t)=1Δk(t)=1 and 2t2−5t+22t2−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IEIE, and there is only one knot with Property IEIE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.