Topological Sliceness of the Figure-Eight Cable

Definitions and mathematical statement

Let $K=4_1\subset S^3=\partial B^4$ be the figure-eight knot, where $B^4$ is the standard closed four-dimensional ball. Let $N(K)$ be a tubular neighborhood of $K$, and let $\mu,\lambda\subset\partial N(K)$ denote a meridian and the preferred longitude, respectively. The preferred longitude has linking number zero with $K$. Use the cabling convention in which the $(2,1)$-cable $C_{2,1}(K)$ is represented by an embedded simple closed curve on $\partial N(K)$ in the homology class \[ 2[\lambda]+[\mu]\in H_1(\partial N(K);{\mathbb{Z}}). \] Thus its longitudinal winding number is two. The supplied record does not explicitly specify its slope convention; this standard convention is made explicit for review.

A knot $J\subset S^3$ is topologically slice if there exists a proper, locally flat topological embedding $f:D^2\hookrightarrow B^4$ such that \[ f^{-1}(\partial B^4)=\partial D^2, \qquad f(\partial D^2)=J. \] Local flatness is required also along the boundary, using the corresponding half-space local models.

Question. Is $C_{2,1}(4_1)$ topologically slice? Equivalently, does it bound a properly embedded, locally flat topological disk in the standard $B^4$?

The supplied dated assessment labels this question open and reports that the cable is not smoothly slice. It further reports that the available methods discussed in the report do not obstruct topological sliceness. These are supplied source qualifications, not independently verified conclusions. The appended smooth-surface argument in the original extraction is corrupted and refers to an unspecified Theorem 5; it does not supply a topological nonsliceness conclusion.

Short English statement

Does the $(2,1)$-cable of the figure-eight knot bound a properly embedded, locally flat topological disk in the standard four-ball? The supplied record labels this question open and distinguishes it from the reported failure of smooth sliceness.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30005375$, title Topological Sliceness of the Figure-Eight Cable, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is 2023; no individual proposer is supplied.
• Low-dimensional Topology (2023), Oberwolfach Reports, pp. 151--213, DOI 10.4171/owr/2023/3. Supplied original report reference: https://doi.org/10.4171/owr/2023/3.
• Supplied literature reference: Ray, topological slice genus of satellite knots, https://arxiv.org/abs/1908.03760.
• Supplied literature reference: Dai--Kang--Mallick--Park--Stoffregen, https://doi.org/10.1007/s00222-024-01286-w. No title or specific result identifier is supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified here. The assessment labels the question open, reports smooth nonsliceness, and states that the report leaves topological sliceness unobstructed by the available methods it discusses. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. This records the supplied verification claim; no source visit or independent verification is asserted here.