Kirby Problem 1.57

Definitions and mathematical statement

Let $K\subset S^3$ be an oriented smooth knot. Its zero-trace $X_0(K)=B^4\cup_K(D^2\times D^2)$ is obtained by attaching a two-handle with the zero framing given by the preferred longitude. Its second integral homology is infinite cyclic, generated by the two-handle core capped by a Seifert surface pushed into $B^4$. The zero-shake genus is the least genus of a closed connected oriented embedded surface representing a generator of $H_2(X_0(K);\mathbb Z)$. Use smooth embeddings for $g^0_{\rm sh}(K)$ and locally flat topological embeddings for $g^{0,\mathrm{TOP}}_{\rm sh}(K)$. A knot is zero-shake slice when the relevant minimum is zero.

Ask separately in both categories: does zero-shake sliceness imply that $K$ bounds a disk in $B^4$ in that category? The additional question is whether some knot satisfies \[ g^{0,\mathrm{TOP}}_{\rm sh}(K)<g^{\mathrm{TOP}}_4(K), \] where $g^{\mathrm{TOP}}_4$ minimizes genus of locally flat oriented surfaces bounded by $K$ in $B^4$. The representative of the trace generator is not required to meet the two-handle cocore in exactly one point.

Short English statement

Does a generator sphere in a knot's zero-trace force a slice disk, and can topological zero-shake genus be smaller than topological slice genus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2716 (KP-1.57), 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 1.57. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.