[#P3082] Cabling conjecture for reducible Dehn surgery

Cable knot surgery producing a reducible manifold. — A structural knot diagram of the statement's mathematical objects.

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?

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

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]

1Records

Record	Kind	Assessment
	Result	Supported
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 →
	Result	Supported
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 →
	Trace	Supported
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 →
	Trace	Reported
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 materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemCabling conjecture for reducible Dehn surgery

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Whitehead asphericity conjecturetopology
Purely cosmetic surgery conjecturetopology
Kashaev volume conjecture for hyperbolic knotstopology

How to cite

TheoremDB contributors, “Cabling conjecture for reducible Dehn surgery,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/cabling-conjecture

This page as plain text: cabling-conjecture.md

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

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

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance