Inverse Kirby knot Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the data we need for the second surgery is just a framed knot $K'$ in $M$, which could have come from another knot $K''$ in $S^3$ before any surgery has been performed. In this case, $K \cup K''$ must be Kirby equivalent to the empty knot because surgery only it is trivial. Therefore, I call $K''$ an inverse Kirby knot of $K$.
Curiously, while the procedure above is straightforward, I don't see an easy way to construct $K''$ directly from a knot presentation of $K$!

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*How to write down a knot presentation of $K''$ from that of $K$?

*Is such $K''$ unique up to Kirby moves?

*What if we start from a link $L$ in $S^3$ instead of a knot?


EDIT According to Danny Ruberman's comment below, the dual knot can be regarded as what I call an inverse Kirby knot.
 A: The standard model for the knot $K''$ you describe is simply the meridian $\mu_K$ of the knot $K$. To be a bit pedantic, this really means that you draw the meridian in the complement of $K$, and then take its image in the surgered manifold. Is this perhaps what you mean by construct $K''$ directly from a knot presentation of $K$?. Since $K''$ is a knot in some $3$-manifold (ie the surgery on $K$) then it's also not clear to me what `knot presentation' means.  The framing on $K''$ to get back to $S^3$ is determined by the pushoff of $\mu$ that links $\mu$ zero times.  The same observations would work for links.
If $Y$ is obtained from $S^3$ by surgery on two distinct knots, say $K_1$ and $K_2$, then each of those would give rise to knots $K_1''$ and $K_2''$ in $Y$ such that surgery along those two knots give back $S^3$. There are many examples of $3$-manifolds with multiple knot surgery descriptions, and presumably the knots $K_1''$ and $K_2''$ would be distinct as knots in $Y$.
The oldest examples I know of manifolds with more than one knot surgery description are due to Brakes, Manifolds with multiple knot-surgery descriptions. Proc. Cambridge Phil. Soc.
87 (1980), 443-448.
