The Weak Gordon–Luecke Conjecture

Definitions and mathematical statement

A closed oriented $3$-manifold is a compact oriented $3$-manifold without boundary. Irreducibility means that every embedded $2$-sphere, in the manifold category intended by the source, bounds an embedded $3$-ball. A knot $K\subset Y$ is an embedded circle. Its complement $Y\setminus K$ carries the orientation induced from $Y$. Two knots are equivalent here if some orientation-preserving self-homeomorphism of $Y$ takes one onto the other; no orientation on the knots themselves is specified.

The supplied standalone formulation of the weak Gordon--Luecke conjecture is the following. For every closed oriented irreducible $3$-manifold $Y$ and every pair of knots $K_1,K_2\subset Y$, neither of which is a rational unknot in a lens space, assert that \[ \begin{aligned} &\exists\text{ an orientation-preserving homeomorphism } h:Y\setminus K_1\longrightarrow Y\setminus K_2\\ &\qquad\Longleftrightarrow \exists\text{ an orientation-preserving homeomorphism } F:Y\longrightarrow Y\text{ with }F(K_1)=K_2. \end{aligned} \] This is an existence statement: it does not require every given complement homeomorphism $h$ to extend over $Y$. The complements displayed here are the complements of the embedded circles, as written in the supplied statement; no boundary parametrization or peripheral marking is specified.

The exceptional phrase "rational unknot in a lens space" is retained verbatim because the supplied record does not define that class or specify its lens-space conventions. The original formulation also refers to "the usual nonexceptional hypotheses" without enumerating them. The standalone version supplies the displayed exception, but the record alone does not establish that it exhausts those hypotheses. Consequently, the intended exceptional class and the complete hypothesis list require source review before this can be regarded as a fully specified mathematical formulation.

Short English statement

The supplied conjecture says that, subject to its exceptional-knot restriction, two knots in the same closed oriented irreducible $3$-manifold have orientation-preservingly homeomorphic complements exactly when an orientation-preserving homeomorphism of the ambient manifold carries one knot to the other. The exceptional class and the original reference to further nonexceptional hypotheses remain insufficiently specified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004814$, title "The Weak Gordon--Luecke Conjecture," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Local registry locations identified by the project instructions are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied report citation: MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/42, pp. 2327--2364. Supplied source URL and literature-source link labelled "OWR report": https://doi.org/10.4171/owr/2021/42. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Additional supplied literature-source link, labelled "Aceto et al.": https://doi.org/10.1112/jlms.70600. No fuller bibliographic citation or specific supporting result is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the complement-determines-knot assertion for knots in general $3$-manifolds is not proved, with the weak Gordon--Luecke formulation tied to still-open cosmetic-surgery questions. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its standalone statement and hypotheses were checked against the official Oberwolfach report. These are supplied verification claims, not verification performed for this draft. The proposed year is recorded as 2021, whereas the report citation gives 2022 and its DOI contains 2021. No individual proposer is named.