Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. However, if I know many surgeries, perhaps I can recover the knot? Let me be specific:
Suppose I have a 2-component link $U_1 \cup U_2$ inside the 3-sphere $S^3$ which has linking number $0$ and such that each component $U_1$, $U_2$ is the unknot.
I'm interested in knowing how much surgery tells us in this situation. If I do $1/n$-surgery on $U_2$ I get back $S^3$, but now $U_1$ sits inside $S^3$ as a knot $K(n)$.
Does the sequence $K(1), K(2), K(3), \ldots$ determine the original link $U_1 \cup U_2$ ? Would it even be expected to?.