Kirby Problem 1.46

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All knots and links below are smooth in the oriented $3$-sphere, up to ambient isotopy, unless another ambient manifold is specified. Let $g_4^{\mathrm{top}}$ and $g_4$ denote the minimum genera of connected oriented locally flat and smooth surfaces in $B^4$ bounding a knot. For a Seifert matrix $V$, its Levine--Tristram signature at $\omega\in S^1\setminus\{1\}$ is the signature of $(1-\omega)V+(1-\bar\omega)V^T$; the maximum below is the maximum absolute value, evaluated away from Alexander roots. A two-bridge knot is a knot admitting a two-bridge sphere, equivalently a Schubert knot $K(p,q)$ with odd $p$ and $\gcd(p,q)=1$. Ribbon means bounding a smooth disk with no radial interior maxima. Topologically homotopy-ribbon means bounding a locally flat disk whose exterior receives a surjection on fundamental groups from the knot exterior. Casson--Gordon ribbon and topological sliceness obstructions refer to the character-signature conditions for prime-power cyclic branched covers in the original Casson--Gordon construction, distinguished as in the cited remarks; their completeness means that passing the respective entire family of conditions suffices for the respective property.

Statement recorded in the supplied source.

(a) Determine $g_4^{\mathrm{top}}(T_{p,q})$ for all torus knots. In particular, is\[g_4^{\mathrm{top}}(T_{p,q})=\tfrac12\max_\omega|\sigma_\omega(T_{p,q})|\,?\] (b) For two-bridge knots, are topological sliceness, ribbonness and topological homotopy-ribbonness equivalent? (c) Are the Casson--Gordon ribbon and topological sliceness obstructions complete for two-bridge knots? (d) Are the differences $g_4(K)-g_4^{\mathrm{top}}(K)$ unbounded among two-bridge knots? [S1][S2]

Short English statement

Determine topological slice genera of torus knots and the effectiveness of ribbon and slice obstructions for two-bridge knots, including the possible size of the smooth versus topological genus gap.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2705 (KP-1.46). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.46 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.