Distinguishing homology $S^1 \times S^2$'s which bound homotopy $S^1$'s Due to Mazur, Akbulut and Kirby  and many others, there are many examples of integer homology 3-spheres which bound contractible 4-manifolds given by attaching a single 2-handle to $S^1 \times D^3$ which algebraically intersects the one-handle once but maybe geometrically many times. Many of these can be distinguished from the 3-sphere by things like the Casson invariant and other invariants specifically developed for integral homology $S^3$'s for instance in https://arxiv.org/pdf/1508.01491.pdf
I would like to know how to distinguish the analogous constructions (which don't reduce to the above case) for homology $S^1 \times S^2$'s bounding homotopy $S^1 \times D^3$'s (homotopy equivalent not homeomorphic). Specifically are there any examples of knots in $S^1\times S^2 \# S^1 \times S^2$ which algebraically intersect both one handles once but not geometrically, that surger to (hopefully many or even infinitely for the application I have in mind) non-trivial examples of homology $S^1 \times S^2$'s? And if so, how does one distinguish these? 
Note by construction these all bound homotopy $S^1 \times D^3$'s. It's very easy to draw such candidates knots, but they end up having a decent amount of crossings, and even then I don't know what tools I would use to distinguish their surgeries from $S^1 \times S^2$ or each other.
 A: A standard construction would be to take a 3-manifold $Y$ given by 0-surgery on a knot $K$. If $K$ is the boundary of a slice disk $D \subset B^4$, then the complement, say $W$ of a neighborhood of $D$ is a homology $S^1 \times B^3$ with boundary $Y$. But you are asking for more, since $W$ should be a homotopy $S^1 \times B^3$; this is equivalent to its fundamental group being ${\bf Z}$. This would require that the Alexander polynomial of $K$ be equal to $1$. 
There are plenty of examples of Alexander polynomial 1 knots bounding a disk with $\pi_1(B^4-D) = \bf{Z}$. Three nice ones are given in the paper Doubly slice knots with low crossing number (New York J. Math. 21 (2015) 1007–1026) of Livingston and Meier. See section 4.2.
If you are willing to accept a topological (rather than smooth) $W$, then Freedman has shown any homology $S^1 \times S^2$ with Alexander polynomial equal to 1 bounds such a $W$. Lots of $S^1 \times S^2$s with Alexander polynomial equal to 1 do not bound a smooth homotopy $S^1 \times B^3$.
