# P3132: Purely cosmetic surgery conjecture

- ID: `P3132`
- Reference: `purely-cosmetic-surgery-conjecture`
- Page: https://theoremdb.org/statements/P3132
- Record maturity: Reviewed problem with recorded work

## Problem

If \(K\subset S^3\) is nontrivial and \(r\ne s\) are two slopes, can the oriented manifolds \(S^3_r(K)\) and \(S^3_s(K)\) ever be orientation-preservingly homeomorphic? The conjecture says no.

### Context

Known frontier: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.

Open boundary: The universal statement for nontrivial knots in S³ remains open.

### Problem setup

- **Definition (slope).** An unoriented primitive isotopy class on the boundary torus of the knot exterior.
- **Definition (purely cosmetic).** Distinct slopes yield orientation-preservingly homeomorphic filled manifolds.
- **Remark.** Distinct fillings can occasionally agree after reversing orientation. Purely cosmetic means the orientation is preserved, which is the rigid case targeted here.

### What counts as a solution

- Prove no nontrivial knot in S³ admits a purely cosmetic pair.
- Or exhibit a nontrivial knot, distinct slopes, and an orientation-preserving homeomorphism of the surgeries.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. Exact unresolved remainder: The universal statement for nontrivial 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: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. Exact unresolved remainder: The universal statement for nontrivial 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: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.

The exact unresolved remainder is: The universal statement for nontrivial knots in S³ remains open.

A complete resolution must meet the following acceptance conditions:
- Prove no nontrivial knot in S³ admits a purely cosmetic pair.
- Or exhibit a nontrivial knot, distinct slopes, and an orientation-preserving homeomorphism of the surgeries.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. Exact unresolved remainder: The universal statement for nontrivial knots in S³ remains open.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: purely cosmetic surgery conjecture open 2026; cosmetic surgery knots S3 distinct slopes 2025
- Strongest checked neighboring result: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.
- Exact unresolved remainder: The universal statement for nontrivial knots in S³ remains open.

### Other known results

- **Claim 2** (supported): Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. [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: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. Unresolved remainder: The universal statement for nontrivial knots in S³ remains open. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The universal statement for nontrivial knots in S³ remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `purely-cosmetic-surgery-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>“Surgery formulae for Casson's invariant and extensions to homology lens spaces.”. Journal für die reine und angewandte Mathematik (Crelles Journal) 1990(405) (1990), 181-220. DOI 10.1515/crll.1990.405.181. cosmetic-surgery consequences https://doi.org/10.1515/crll.1990.405.181
   - Also cited at S. Boyer and D. Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990). cosmetic-surgery consequences
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides early invariant constraints on cosmetic surgeries.
   - Source used to assess the problem's recorded status.
   - For Purely cosmetic surgery conjecture: This is the dated publication status for the canonical target Purely cosmetic surgery conjecture.
   - Source named by the research packet.
2. <a id="reference-2"></a>Kazuhiro Ichihara and In Dae Jong, “Large alternating Montesinos knots do not admit purely cosmetic surgeries”. Topology and its Applications 371 (2025), 109354. DOI 10.1016/j.topol.2025.109354. main theorem https://doi.org/10.1016/j.topol.2025.109354
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves the conjecture for broad alternating and Montesinos families while stating the general form.
   - Source used to assess the problem's recorded status.
   - For Purely cosmetic surgery conjecture: Proves the conjecture for broad alternating and Montesinos families while stating the general form.