[#P3132] Purely cosmetic surgery conjecture

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.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

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.

2Problem setup

Definition 1 (slope). An unoriented primitive isotopy class on the boundary torus of the knot exterior.

Definition 2 (purely cosmetic). Distinct slopes yield orientation-preservingly homeomorphic filled manifolds.

Remark 1. Distinct fillings can occasionally agree after reversing orientation. Purely cosmetic means the orientation is preserved, which is the rigid case targeted here.

3What 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.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (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.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	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: 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][2] Relevance to this problem This is the dated publication status for the canonical target Purely cosmetic surgery conjecture. 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: 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. Open full record →
Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.^[1][2] Relevance to this problem Locates the present research frontier immediately below Purely cosmetic surgery conjecture. 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 Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. This leaves the following boundary unresolved: The universal statement for nontrivial 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: 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.	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: 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][2] Relevance to this problem Documents why Purely cosmetic surgery conjecture 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: - purely cosmetic surgery conjecture open 2026 - cosmetic surgery knots S3 distinct slopes 2025 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. Related: Claim 1 Open full record →
The universal statement for nontrivial knots in S³ remains open.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The universal statement for nontrivial knots in S³ remains open. Relevance to this problem Turns the remaining uncertainty in Purely cosmetic surgery conjecture 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 universal statement for nontrivial knots in S³ remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - 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. 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: 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.

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.

How the 4 records connect

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

ProblemPurely cosmetic surgery conjecture

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

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Purely cosmetic surgery conjecture.” TheoremDB. P3132. Problem statement; statement text SHA-256 caffb6c7ab3497241869a8c6a2cd3fe2509cf2c78cc24f2238c8076a3cfa3750. https://theoremdb.org/statement/?ref=P3132

BibTeX

@misc{theoremdb-problem-caffb6c7ab3497241869a8c6a2cd3fe2509cf2c78cc24f2238c8076a3cfa3750,
  title = {{Purely cosmetic surgery conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 caffb6c7ab3497241869a8c6a2cd3fe2509cf2c78cc24f2238c8076a3cfa3750},
  url = {https://theoremdb.org/statement/?ref=P3132}
}

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. “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. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Provides early invariant constraints on cosmetic surgeries.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.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.
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. ↗journal article · primary source · checked 2026-08-01Source 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.

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

[#P3132] Purely cosmetic surgery conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance