Kirby Problem 1.47

Definitions and mathematical statement

For a smooth knot $K\subset\partial B^4$, consider compact, connected, nonorientable smooth surfaces $F$ properly embedded in $B^4$ with boundary $K$. Write $b_1(F)=\dim H_1(F;\mathbb Q)$ and define \[ \gamma_4(K)=\min_F b_1(F). \] Here the competitors are nonorientable, including when $K$ also bounds an orientable surface. The normal Euler number $e(F)\in\mathbb Z$ is the relative Euler number of the normal bundle, evaluated with orientation local coefficients, using the zero-linking boundary framing of $K$. It measures the obstruction to extending that boundary normal section to a nowhere-zero normal section on $F$.

Determine $\gamma_4(K)$ and, more precisely, the geography set \[ \mathcal G(K)=\{(e(F),b_1(F)):F\text{ as above}\}\subset\mathbb Z\times\mathbb N. \] The designated family question asks for these minimal first Betti numbers for torus knots $T_{p,q}$, where $p,q$ are coprime integers of absolute value greater than one and $T_{p,q}$ winds in the two directions on the standard torus in $S^3$. A geography determination must specify exactly which pairs are realizable, as well as obstructions to all excluded pairs.

Short English statement

Determine the least complexity of a nonorientable surface bounded by a knot, and all possible complexities and normal Euler numbers.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2706 (KP-1.47), 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 1.47. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.