Kirby Problem 1.84

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.

For a classical knot $K\subset S^3$, the smooth slice genus $g_4^{\mathrm{sm}}(K)$ is the least genus of a connected smooth oriented surface properly embedded in $B^4$ with boundary $K$; $g_4^{\mathrm{top}}$ uses locally flat topological embeddings. A virtual knot is represented by an embedding in $\Sigma\times[0,1]$, for a compact oriented surface $\Sigma$, with representations identified under isotopy and addition or removal of handles away from the knot. Its virtual slice genus $g_v$ is the least genus of a connected oriented properly embedded surface $F\subset M\times[0,1]$ with boundary the represented knot in $\partial M\times[0,1]$, where $M$ ranges over compact oriented 3-manifolds with $\partial M=\Sigma$, and representatives range over the virtual equivalence class. Again distinguish smooth and locally flat versions. A classical knot gives a virtual knot by viewing it in a thickened sphere.

Statement recorded in the supplied source.

For every classical knot $K$, do both equalities hold? \[g_v^{\mathrm{sm}}(K)=g_4^{\mathrm{sm}}(K),\qquad g_v^{\mathrm{top}}(K)=g_4^{\mathrm{top}}(K).\] The two categories are separate versions of the question. [S2]

Short English statement

Can allowing virtual ambient surfaces lower the slice genus of a classical knot, either smoothly or in the locally flat topological category?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2743 (KP-1.84). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.84 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.