# Construction of Kirby-diagram for slice-complement

Let $(S^3,K)=\partial (D^4,D)$ be a slice knot. (i.e. a knot in $S^3$ bounds a disk in $D^4$.)

Are there any relation between the knot diagram of $K$ and the kirby diagram for $(D^4-D,M_K)$, where $M_K$ is the 3-manifold obtained by 0-surgery on $S^3$ along $K$?

• The complement of the slice disc can require arbitrarily complicated handle presentations, depending roughly on how "knotted" the slice disc is. For example, given a slice disc for the trivial knot, you can cap it off to get a knotted $S^2$ in $S^4$, and this is a reversible process, all $2$-knots are capped-off slice discs for the trivial knot. In particular the complement if this 2-knot is difeomorphic to the complement of the corresponding slice disc. So that gives you a sense for how complicated it can get. – Ryan Budney May 15 '11 at 19:01

As Ryan Budney points out in a comment, there's not much you can say without specifying the slice disk $D$. If you have an explicit presentation of $D$ as a ribbon disk, then you can easily turn this into an explicit handle diagram for $D^4 \setminus D$. If you cut the ribbon self-intersections of the ribbon disk, the result is an unlink in $S^3$, and this unlink gives the 1-handles of the diagram (in Kirby notation, dotted circles). For each ribbon, add a 0-framed 2-handle which links each of the adjacent dotted circles.
If you have a slice disk $D$ which is not a ribbon disk, then the associated Morse function on $D$ will have local maxima, and each such maximum will correspond to a 3-handle in a generalization of the above recipe for a handle diagram.