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 . 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 Δ k ( t ) must be either 1 or 2 t 2 − 5 t + 2 , and we give two infinite families of non-fibred genus 1 knots with Property I E and having Δ k ( t ) = 1 and 2 t 2 − 5 t + 2 respectively. 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. o give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.