[#P3132] Purely cosmetic surgery conjecture

Two distinct Dehn fillings compared for oriented homeomorphism. — A structural knot diagram of the statement's mathematical objects.

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.

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

Current 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]

1Records

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

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 connectTyped relations and evidence flow

ProblemPurely cosmetic surgery conjecture

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
Kashaev volume conjecture for hyperbolic knotstopology
Dürer’s edge-unfolding problemtopology

How to cite

TheoremDB contributors, “Purely cosmetic surgery conjecture,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/purely-cosmetic-surgery-conjecture

This page as plain text: purely-cosmetic-surgery-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. “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

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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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