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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 $K$?

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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.

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It would be wonderful if there were bounds on the complexity of a slice disc for a slice knot, in terms of some kind of measure of the complexity of a knot -- crossing-number, or perhaps the number of tetrahedra it takes to triangulate the complement.

The kind of bound I'd really like to have is how many times you have to refine the triangulation of the 4-ball for the slice disc to appear as a "normal surface" in the triangulated 4-ball. That, or something similar to that could allow for computer-searches for slice knots that aren't known to be ribbon knots.

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