Slice Candidates and Shared Trace Boundaries of Knots

Definitions and mathematical statement

Under the usual smooth-knot interpretation, a knot is a smooth embedding of $S^1$ in $S^3=\partial B^4$, considered up to ambient isotopy. A knot $K$ is smoothly slice if there is a smooth proper embedding $d\colon D^2\hookrightarrow B^4$ whose boundary is $K$. The source questions say only "slice"; the supplied literature assessment specifically concerns smooth sliceness. The statements below use smooth sliceness, subject to confirmation of the intended category.

Let $T_-$ be the left-handed trefoil and let $4_1$ be the figure-eight knot. Write $D_{+,t}(T_-)$ for the Whitehead satellite of $T_-$ with positive clasp and integer twisting parameter $t$, measured relative to the Seifert framing; $t=0$ denotes the untwisted pattern. The source specifies a positive Whitehead double but does not explicitly specify its twisting. Let $C_{2,1}(4_1)$ denote the cable represented on the boundary of a tubular neighborhood of $4_1$ by the slope $2\lambda+\mu$, where $\lambda$ is the preferred Seifert longitude and $\mu$ is a meridian. Thus the cable winds twice longitudinally and once meridionally.

For $n\in{\mathbb{Z}}$, define the $n$-framed knot trace by \[ X_n(K)=B^4\cup_{K,n}(D^2\times D^2),\qquad Y_n(K)=\partial X_n(K), \] where a smooth $2$-handle is attached along $K$ with framing $n$ relative to the Seifert framing. Give $Y_n(K)$ the boundary orientation. The source writes $X(K)$ without defining its framing. The common interpretation $X(K)=X_0(K)$ therefore remains a convention to confirm.

For closed oriented smooth $3$-manifolds $Y_0,Y_1$, write $Y_0\sim_{{\mathbb{Z}} H}Y_1$ if there is a compact oriented smooth $4$-manifold $W$ with \[ \partial W=(-Y_0)\sqcup Y_1 \] and both inclusion maps induce isomorphisms \[ H_i(Y_j;{\mathbb{Z}})\xrightarrow{\cong}H_i(W;{\mathbb{Z}}) \quad\text{for every }i\geq 0\text{ and }j\in\{0,1\}. \] This is the smooth oriented interpretation of integer-homology cobordism. The record does not explicitly specify the category or orientation convention for this relation, or whether its boundary diffeomorphisms must preserve orientation.

With these qualifications, the questions are:

1. Is the positive Whitehead double $D_{+,t}(T_-)$ named in the source smoothly slice? Here $t$ is the source's intended twisting parameter, not a universally quantified parameter; under the untwisted convention this is the question for $t=0$.
2. Is $C_{2,1}(4_1)$ smoothly slice?
3. Let $\mathcal K$ denote the ambient-isotopy classes of smooth unoriented knots in $S^3$. For the framing $n$ intended by $X(K)$, determine the sets \[ \mathcal A_n=\left\{[K]\in\mathcal K:\exists[J]\in\mathcal K,\ [J]\neq[K],\ Y_n(J)\cong Y_n(K)\right\}, \] \[ \mathcal B_n=\left\{[K]\in\mathcal K:\exists[J]\in\mathcal K,\ [J]\neq[K],\ Y_n(J)\sim_{{\mathbb{Z}} H}Y_n(K)\right\}. \] Here $\cong$ denotes a smooth diffeomorphism, with its orientation requirement unresolved. The parameter $n$ records missing source notation; the source does not explicitly ask for a classification for every framing. If $X(K)$ means the zero trace, the requested sets are $\mathcal A_0$ and $\mathcal B_0$. Ambient nonisotopy is the working meaning of "distinct."
4. Give an algorithm for producing knots $J$ satisfying the requested relation with a given knot $K$. The source does not specify an input encoding, which of the two relations the algorithm must handle, or its required behavior when no such $J$ exists; consequently no precise termination or decision requirement can be extracted.

The word "merely" is interpreted above as allowing the weaker relation of integer-homology cobordism. If it requires the boundaries additionally to be nondiffeomorphic, the definition of $\mathcal B_n$ must include that extra condition; the supplied wording does not settle this.

The supplied dated curation labels the combined record partially solved. It reports that the $(2,1)$-cable of the figure-eight knot has been proved not smoothly slice, while the positive Whitehead-double question and the general trace-boundary algorithm problem remain open. These are attributed source assessments, not independently verified conclusions of this draft.

Short English statement

The record asks whether two specified satellite knots are slice, which knots share a trace boundary or an integer-homology-cobordant trace boundary with a distinct knot, and how to construct such partners algorithmically. The supplied curation reports a smooth nonsliceness result for the cable. The Whitehead twisting, trace framing, category and orientation conventions, and algorithmic requirements need confirmation before the entire statement is unambiguous.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004412$, exact source problem code catalogue entry, titled "Slice Candidates and Shared Trace Boundaries of Knots," in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2020; no individual proposer is supplied.
• Supplied report citation: Low-dimensional Topology (2021), Oberwolfach Reports, pp. 465--516, DOI 10.4171/owr/2020/8. https://doi.org/10.4171/owr/2020/8. The literature-source list also labels this reference "Oberwolfach Report 8/2020." The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Dai, Hom, Stoffregen, and Truong, supplied reference label "cable-slice obstruction." https://arxiv.org/abs/2207.14187.
• Weber, supplied reference label "notes on the Oberwolfach knot problems." https://numdam.org/item/10.5802/wbln.39.pdf.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the record partially solved, reports that the $(2,1)$-cable of the figure-eight knot is not smoothly slice, and describes the positive Whitehead-double question and the general trace-boundary algorithm problem as open. The supplied background explicitly requires independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and states that the official report was checked and that both sliceness questions, the trace-boundary questions, and an omitted algorithm request were restored. This is a supplied verification claim; no source visit or independent verification is asserted here.