Kirby Problem 1.52

Definitions and mathematical statement

Encode a smooth knot $K\subset S^3$ by a finite regular diagram, for example a plane four-valent graph with crossing information and its single traversal. The input is promised to be a valid knot diagram. The knot is smoothly slice if it bounds a properly embedded smooth disk in the standard $B^4$. It is ribbon if such a disk can be isotoped so the radial function has no interior local maxima; a disk-and-band description is an equivalent certificate.

For each property separately, is there a terminating algorithm that takes every knot diagram and returns yes exactly when the represented knot has that property? Thus the two requested decision procedures concern sliceness and ribbonness, respectively. Their answers must be invariant under diagram moves that leave the knot type unchanged.

No polynomial running-time bound is stipulated. Nevertheless, an unbounded search that eventually finds a ribbon presentation in positive cases alone is not a decision procedure: it must also halt and return no on every negative instance. The ambient four-ball and the smooth category are fixed in both questions.

Short English statement

Can an algorithm always decide whether an input knot is smoothly slice, or whether it is ribbon?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2711 (KP-1.52), 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 1.52. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.