Kirby Problem 1.41

Definitions and mathematical statement

For a smooth oriented knot $K\subset S^3=\partial B^4$, let $g_4(K)$ be the least genus of a smooth connected oriented surface properly embedded in $B^4$ with boundary $K$. Write $nK$ for the connected sum of $n$ copies. Subadditivity of $g_4$ makes the following limit exist: \[ g_s(K)=\lim_{n\to\infty}\frac{g_4(nK)}n =\inf_{n\geq1}\frac{g_4(nK)}n. \] The smooth concordance group identifies knots cobounding smooth annuli in $S^3\times[0,1]$; its operation is connected sum. A class $[K]$ is torsion if $mK$ bounds a smooth disk in $B^4$ for some $m\geq1$.

The four questions are:

1. Can $g_s(K)$ be rational but not an integer?
2. Can $g_s(K)$ be irrational?
3. Can $0<g_s(K)<1/2$?
4. Is $g_s(K)=0$ equivalent to $[K]$ being torsion?

All genera and concordances here are smooth. Torsion already forces stable genus zero by subadditivity; the reverse implication in (d) asks whether sublinear growth can occur for a class of infinite order.

Short English statement

What values can stable smooth four-genus take, and does it vanish exactly on torsion concordance classes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2700 (KP-1.41), 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 1.41. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.