Kirby Problem 1.59

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.

A slice disk for $K\subset S^3$ is a smooth properly embedded disk $D\subset B^4$ with $\partial D=K$. A ribbon disk has no radial local maxima after isotopy. A homotopy-ribbon disk has surjective inclusion map $\pi_1(S^3\setminus\nu K)\to\pi_1(B^4\setminus\nu D)$. Isotopies of disks here are not required to fix the boundary pointwise. Local knotting means interior connected sum of the disk with a smoothly embedded 2-sphere in $S^4$.

Let $\mathcal R(K)$ be ribbon disks modulo isotopy, $\mathcal D(K)$ all slice disks modulo isotopy and local knotting, and $\mathcal{HR}(K)$ homotopy-ribbon disks modulo isotopy. For fibered $K$, let $\mathcal{FHR}(K)$ consist of homotopy-ribbon disks in smooth homotopy 4-balls bounded by $S^3$ whose exteriors fiber over $S^1$ with handlebody fibers inducing the knot fibration. Equivalence includes identifications of the ambient pairs and disk isotopy, without a pointwise boundary constraint.

Statement recorded in the supplied source.

For every smooth slice knot $K\subset S^3$, determine the sets $\mathcal R(K)$, $\mathcal D(K)$ and $\mathcal{HR}(K)$. For fibered slice knots also determine $\mathcal{FHR}(K)$. For the natural map $\iota_K:\mathcal{HR}(K)\to\mathcal D(K)$, investigate which knots make it injective, surjective, or have finite image. [S2]

Short English statement

Classify ribbon disks, slice disks modulo local knotting, homotopy-ribbon disks, and fibered homotopy-ribbon disks for a fixed boundary knot; also understand the natural map forgetting the homotopy-ribbon condition.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2718 (KP-1.59). 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 1.59 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.