Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane.
The question is then how many spin structures there are in $M$, for a given knot $K$? Since $M$ has a boundary $\partial M=T^2$, and since $T^2$ has four spin structures, I think I could refine this question by asking, "how many spin structures are there in $M$, which extends a fixed spin structure (one of the four) on $\partial M$?
I believe this is related to the computation of $H^1(M, \partial M; \mathbb{Z}_2)$, but do not know how to compute this.