Kirby Problem 1.50

Definitions and mathematical statement

Work smoothly in the standard ball $B^4$ with boundary $S^3$. A knot is slice if it bounds a properly embedded disk in $B^4$. An $m$-component link is slice if its components bound $m$ pairwise disjoint such disks. A properly embedded surface is ribbon if, after isotopy relative to the boundary, the radial function on its interior is Morse with no local maxima. Equivalently it has a presentation by disks and bands, with only index-zero and index-one critical points. A ribbon knot or link bounds disks with this property. For a knot $K$, its smooth four-genus $g_4(K)$ is the least genus of a connected oriented smooth surface in $B^4$ bounded by $K$.

The questions are:

1. Does every slice knot bound a ribbon disk?
2. Does every slice link bound a disjoint collection of ribbon disks?
3. Does every knot bound a connected oriented ribbon surface of genus exactly $g_4(K)$?

In (b), the disks must be disjoint simultaneously; sliceness of each component considered by itself is a weaker hypothesis. Part (c) specializes to (a) when the four-genus is zero.

Short English statement

Can slice disks always be chosen ribbon, and can every minimal-genus surface in the four-ball be replaced by a ribbon surface of the same genus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2709 (KP-1.50), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.50. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.