Let $X$ be a smooth closed connected 4-manifold. It admits a handlebody structure, having a unique 0- and a unique 4-handle. We can express the handlebody structure as a Kirby diagram (https://en.wikipedia.org/wiki/Kirby_calculus). Suppose $Y$ is a subhandlebody of $X$, which contains the 0-handle of $X$ and some 1,2,3-handles of $X$. (So $Y$ has nonempty boundary.) Then we can view the handlebody structure of $Y$ inside the Kirby diagram of $X$. Is there a method or algorithm to draw the Kirby diagram of the complement $X-\operatorname{int}(Y)$? If not, how about in the simpler case where $X$ has no 1- or 3-handles, so that the Kirby diagram of $X$ is just a link in $S^3$ with integer framings?