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The purely cosmetic surgery conjecture for knots in the three-sphere

For a knot K⊂S3K\subset S^3 and a slope r∈Q∪{∞}r\in\mathbb{Q}\cup\{\infty\}, let Sr3(K)S^3_r(K) be the oriented manifold obtained by Dehn surgery. Two distinct slopes r≠sr\ne s are purely cosmetic if Sr3(K)S^3_r(K) and Ss3(K)S^3_s(K) are orientation-preservingly homeomorphic. Gordon conjectured (as part of his cosmetic filling conjecture) that this never happens for a nontrivial knot; the question is Bleiler's Problem 1.81(A) in Kirby's list. Known before this work: the meridional case (Gordon-Luecke), same-sign pairs (Wu), the reduction to opposite slopes ±2\pm 2 or ±1/q\pm 1/q (Ni-Wu, Hanselman), exclusion of ±1/q\pm 1/q (Daemi-Lidman-Miller Eismeier), and verification for many families and all knots up to about 19 crossings. Does every nontrivial knot K⊂S3K\subset S^3 satisfy: Sr3(K)≅Ss3(K)S^3_r(K)\cong S^3_s(K) orientation-preservingly implies r=sr=s?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Low-dimensional topology: Dehn surgery on knots
Posed by
C. McA. Gordon, Dehn surgery on knots, Proc. ICM Kyoto 1990 (published 1991), Conjecture 6.1; recorded as Bleiler's Problem 1.81(A) in Kirby's 1997 problem list
Year posed
1990
Years open
36y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every nontrivial smooth knot K⊂S3K\subset S^3 and all r,s∈Q∪{∞}r,s\in\mathbb{Q}\cup\{\infty\}, an orientation-preserving homeomorphism Sr3(K)≅Ss3(K)S^3_r(K)\cong S^3_s(K) forces r=sr=s; the homeomorphism need not preserve the surgery core or the knot exterior. After the cited reduction to g(K)=2g(K)=2 and slopes ±2\pm 2, the paper derives a contradiction by comparing a nonzero integer instanton count Ω\Omega on a parameterized cobordism with a one-dimensional count of solutions with spinors whose boundary would force 2ηΩ=02^\eta\Omega=0. It does not address chirally cosmetic (orientation-reversing) surgeries or knots in other three-manifolds.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated by Gordon and in Kirby's list; the statement matches the full conjecture, including the meridional and zero slopes. The proof was not refereed. The argument takes as an input the reduction (Proposition 1.2) that any purely cosmetic pair must be the slopes ±2\pm 2 on a genus-two knot, which the manuscript draws from Daemi, Lidman and Miller Eismeier (arXiv:2410.21248) combined with Hanselman and Ni-Wu; the new work rules out this last case with an instanton count compared against a nonabelian (SO(3)) monopole cobordism in the Pidstrigach-Tyurin, Okonek-Teleman and Feehan-Leness tradition, an area with a history of delicate analytic gaps. No Lean formalization is supplied for this family.

Sources

Changelog1 change

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