  We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if
the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$.
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 $\Delta_k(t)$
must be either 1 or $2t^2-5t+2$, and we give two infinite families of
non-fibred genus 1 knots with Property $IE$ and having $\Delta_k(t)=1$ and
$2t^2-5t+2$ respectively.
  Hence among genus one 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.
