[#P3082] Cabling conjecture for reducible Dehn surgery

Problem. If a nontrivial knot \(K\subset S^3\) has a Dehn surgery producing a reducible three-manifold, must \(K\) be a cable knot and must the surgery slope be its cabling slope?

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Definitions and notation

1Context

Known frontier: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Open boundary: The classification for arbitrary knots in S³ remains open.

2Problem setup

Definition 1 (reducible). Contains an embedded two-sphere that does not bound a three-ball.

Definition 2 (cable knot). A knot lying as a nontrivial torus-knot slope on the boundary of a companion's tubular neighborhood.

Remark 1. A cable knot has an obvious reducible surgery that splits off a lens-space summand. The conjecture says this construction accounts for every reducible surgery on a nontrivial knot in S³.

3What counts as a solution

Prove the cable-and-slope conclusion for every nontrivial knot in S³.
Or give a noncable knot and slope whose surgery is reducible.

1Status

What counts as a solution

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Cabling conjecture for reducible Dehn surgery. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. The exact unresolved remainder is: The classification for arbitrary knots in S³ remains open. A complete resolution must meet the following acceptance conditions: - Prove the cable-and-slope conclusion for every nontrivial knot in S³. - Or give a noncable knot and slope whose surgery is reducible. Open full record →
The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general.^[1][2] Relevance to this problem Locates the present research frontier immediately below Cabling conjecture for reducible Dehn surgery. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. This leaves the following boundary unresolved: The classification for arbitrary knots in S³ remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Unresolved remainder: The classification for arbitrary knots in S³ remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Unresolved remainder: The classification for arbitrary knots in S³ remains open.^[1][2] Relevance to this problem Documents why Cabling conjecture for reducible Dehn surgery was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Cabling Conjecture still open 2025 reducible surgery - thin knots cabling conjecture 2025 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Unresolved remainder: The classification for arbitrary knots in S³ remains open. Related: Claim 1 Open full record →
The classification for arbitrary knots in S³ remains open.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The classification for arbitrary knots in S³ remains open. Relevance to this problem Turns the remaining uncertainty in Cabling conjecture for reducible Dehn surgery into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: The classification for arbitrary knots in S³ remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove the cable-and-slope conclusion for every nontrivial knot in S³. - Or give a noncable knot and slope whose surgery is reducible. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.

Equivalent-formulation queries: Cabling Conjecture still open 2025 reducible surgery; thin knots cabling conjecture 2025
Strongest checked neighboring result: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general.
Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemCabling conjecture for reducible Dehn surgery

All 3 recorded relations between these records and the problem

2See also

Torsion-freeness for king-grid independence complexestopology
Integral torsion in scale-four hypercube Rips complexestopology
Hodge conjecturetopology

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Plain text

“Cabling conjecture for reducible Dehn surgery.” TheoremDB. P3082. Problem statement; statement text SHA-256 c08246dc1edb10bc20e860e930faa9d33054a56eccbe26ac77faec2ba0f44804. https://theoremdb.org/statement/?ref=P3082

BibTeX

@misc{theoremdb-problem-c08246dc1edb10bc20e860e930faa9d33054a56eccbe26ac77faec2ba0f44804,
  title = {{Cabling conjecture for reducible Dehn surgery}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c08246dc1edb10bc20e860e930faa9d33054a56eccbe26ac77faec2ba0f44804},
  url = {https://theoremdb.org/statement/?ref=P3082}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. C.McA. Gordon and J. Luecke, “Reducible manifolds and Dehn surgery”. Topology 35(2) (1996), 385-409. DOI 10.1016/0040-9383(95)00016-X. main reducible-surgery theorems. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Establishes foundational restrictions on reducible Dehn surgeries on knots in S³.Also cited at C. McA. Gordon and J. Luecke, “Reducible manifolds and Dehn surgery,” Topology 35(2) (1996), 385–409. main reducible-surgery theorems.Source used to assess the problem's recorded status.For Cabling conjecture for reducible Dehn surgery: This is the dated publication status for the canonical target Cabling conjecture for reducible Dehn surgery.Source named by the research packet.
Thin knots and the cabling conjecture, Algebraic & Geometric Topology 25 (2025). introduction and main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves new cases while explicitly treating the general conjecture as open.Source used to assess the problem's recorded status.For Cabling conjecture for reducible Dehn surgery: Proves new cases while explicitly treating the general conjecture as open.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3082] Cabling conjecture for reducible Dehn surgery

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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Record & provenance