Figure-Dependent Slice Curves in Disc Complements

Definitions and mathematical statement

Formulation limitation. The supplied record states that the knot $K$ and curves $\alpha,\beta$ are specified only by a figure absent from the dataset. It labels the statement unrecoverable. Consequently, their embeddings and the intended ambient space cannot be recovered from the supplied evidence, and a self-contained mathematical problem cannot be certified.

The retained question asks whether, for every smooth disc $D$ with boundary $K$, at least one of $\alpha$ and $\beta$ slices in the complement of $D$.

The following gives only its conditional logical structure. Let $M$ denote the unspecified intended ambient smooth manifold, and let $K,\alpha,\beta$ denote the unspecified embedded curves from the missing figure. Write \[ \mathbb D^2=\{(x,y)\in{\mathbb{R}}^2:x^2+y^2\leq 1\}, \qquad \mathcal D_M(K)=\{D\subseteq M:D\text{ is a smoothly embedded copy of }\mathbb D^2,\ \partial D=K\}. \] If slicing in the complement means bounding a smoothly embedded disc disjoint from $D$, the question has the schematic form \[ \forall D\in\mathcal D_M(K),\quad \exists\gamma\in\{\alpha,\beta\}\ \exists E\subseteq M\setminus D \quad \left(E\text{ is a smoothly embedded copy of }\mathbb D^2\ \text{and}\ \partial E=\gamma\right). \] Here $\partial D$ and $\partial E$ mean the intrinsic boundaries of the embedded discs. The choice between $\alpha$ and $\beta$ may depend on $D$. The formula does not quantify over arbitrary choices of $M,K,\alpha,\beta$: these are intended to be fixed by the missing source context. The smooth interpretation of slicing, properness and boundary conditions for the discs, and the meaning of complement require confirmation. Thus this schematic formula is not a recovered precise statement of the figure-dependent problem.

Short English statement

For the particular knot and two curves in a missing source figure, the question asks whether every smooth disc bounded by the knot leaves at least one of the two curves slice in its complement. The missing configuration and unspecified ambient and boundary conventions prevent a self-contained formulation. The supplied registry labels the problem open; this status is supplied curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005378$, exact source problem code catalogue entry, title Figure-Dependent Slice Curves in Disc Complements. The record gives proposed year 2023 and names no proposer. Its supplied statement-verification note reports that the official report defines the knot and curves only through a figure absent from the dataset, and therefore marks the statement unrecoverable; this verification claim has not been independently checked here.
• Low-dimensional Topology (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/3, pp. 151--213. Supplied source report and literature-source label: Oberwolfach Report 3/2023. https://doi.org/10.4171/owr/2023/3.
• Gabai, Knotted 3-balls in S4. Supplied literature-source link: https://arxiv.org/abs/1912.09029.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and states that Gabai's related constructions and the Oberwolfach report provide context but no proof or counterexample. This is supplied dated curation derived from the OWR material, not independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.