# P3082: Cabling conjecture for reducible Dehn surgery

- ID: `P3082`
- Reference: `cabling-conjecture`
- Page: https://theoremdb.org/statements/P3082
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (reducible).** Contains an embedded two-sphere that does not bound a three-ball.
- **Definition (cable knot).** A knot lying as a nontrivial torus-knot slope on the boundary of a companion's tubular neighborhood.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The classification for arbitrary knots in S³ remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `cabling-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.1016/0040-9383(95)00016-X
   - Also cited at C. McA. Gordon and J. Luecke, “Reducible manifolds and Dehn surgery,” Topology 35(2) (1996), 385–409. main reducible-surgery theorems
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Establishes foundational restrictions on reducible Dehn surgeries on knots in S³.
   - 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.
2. <a id="reference-2"></a>Thin knots and the cabling conjecture, Algebraic & Geometric Topology 25 (2025). introduction and main theorem https://msp.org/agt/2025/25-8/agt-v25-n8-p03-p.pdf
   - journal_article; primary source; checked 2026-08-01
   - Source 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.