The purely cosmetic surgery conjecture
From OpenAI Aftermath
The purely cosmetic surgery conjecture asks whether two distinct Dehn surgeries on a nontrivial knot in the three-sphere can produce orientation-preservingly homeomorphic three-manifolds. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026, claims that they cannot. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The question is an instance of the cosmetic filling problem formulated by Gordon and recorded as Bleiler's Problem 1.81(A) in Kirby's problem list. Gordon and Luecke's theorem that nontrivial surgery on a nontrivial knot cannot produce the three-sphere settles the meridional case.
- Obstructions to a cosmetic pair of slopes came from classical invariants (Boyer and Lines), from Heegaard Floer homology (Ozsváth and Szabó, Wang, Wu, Ni and Wu) and from immersed curves (Hanselman, who restricted the slopes to ±2 or ±1/q and showed that ±2 requires genus two). Ichihara and Wu used the Jones polynomial, and Detcherry gave a quantum obstruction.
- Special classes were settled by Tao (cables and composite knots) and by Stipsicz and Szabó (pretzel knots); Kotelskiy, Lidman, Moore, Watson and Zibrowius proved an equivariant version for strongly invertible knots. Futer, Purcell and Schleimer verified the conjecture through 19 crossings and for the one-cusped SnapPy census, and Ren reduced it to hyperbolic surgeries on hyperbolic knots.
- The filtered instanton argument of Daemi, Lidman and Miller Eismeier excludes the slopes ±1/q; with the earlier restrictions, only the slopes −2 and 2 on a genus-two knot remain.
The original result
- Claim. Theorem 1.1 of Purely cosmetic surgery on knots in the three-sphere states that if K is a smooth nontrivial knot in S3 and r, s are slopes in Q ∪ {∞} with an orientation-preserving homeomorphism between the surgeries S3r(K) and S3s(K), then r = s. The manuscript stresses that the homeomorphism need not preserve the surgery core or the knot exterior, and that orientation-reversing coincidences are a different question.
- Method, as described: after the reduction to the slopes −2 and 2 on a genus-two knot, the manuscript constructs two counts on a closed parameterized cobordism: a nonzero integer instanton count, and an oriented one-dimensional count of spinor solutions whose boundary would force a power of two times the instanton count to vanish. The comparison follows the nonabelian-monopole cobordism strategy of Pidstrigach–Tyurin and Okonek–Teleman, with the analytic and link methods of Feehan and Leness; symplectic caps and Donaldson invariants, following Kronheimer and Mrowka, give the nonvanishing.
- Source (openai/math repository, commit
7ef5c0c): Purely cosmetic surgery on knots in the three-sphere. - Formalization. The collection lists no Lean formalization for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 28 October 2026; AI Proved What? row 28 October 2026) |
|---|---|---|---|---|
| 2026-09-26 | K. Ichihara, Purely cosmetic surgeries on knots with low-span Jones polynomials, arXiv:2609.32507, v1 | Parallel: a special case by independent methods | Shows that a knot whose nontrivial Jones polynomial has span at most 11 admits no purely cosmetic surgery. | arXiv preprint, posted before the public release of the OpenAI collection on 6 October 2026; the abstract does not mention the OpenAI manuscript. |
| 2026-10-07 | K. Ichihara and K. Kishimoto, Purely cosmetic surgeries on knots with low-span Jones polynomials, II, arXiv:2610.10019, v1 | Parallel: a special case by independent methods | Extends the exclusion to Jones polynomials of span at most 14, determines the Laurent polynomials of span 15 that satisfy the conditions used, an infinite family, and shows that knots of braid index at most 4 whose Jones polynomial has odd span admit no purely cosmetic surgery. | arXiv preprint; the abstract does not mention the OpenAI manuscript. |
| 2026-10-07 | Cosmetic knot surgery: repetition and an oriented boundary count, an article of the Latent Seminar blog at lowrank.github.io | Reading, exposition | Explains the manuscript's argument: the imported reduction to the slopes −2 and 2, the nonzero instanton count obtained by finite repetition, the spinor count with its doubled projective link, the ends that must be excluded and the oriented boundary contradiction. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-10 | F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, OSVerify, Stanford University: referee report on Purely cosmetic surgery on knots in the three-sphere | Review, referee report | Reviews the manuscript version of 23 September 2026 and reports no major issue and three minor issues, each with a proposed repair: the single-interval case misstates how many parameters disappear, an index equality counts virtually restored lens parameters, and a boundary-flux pairing has its sign reversed. | Web page at osverify.github.io, first posted 10 October 2026, last changed 12 October 2026. The page states that its findings and severity labels come from OSVerify's internal review and do not necessarily establish that the main theorem is false; the site's home page states that every reported issue was verified by human experts. |
| 2026-10-18 | AI Proved What?, episode 306 Never the same twice?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim that distinct Dehn surgery slopes on a nontrivial knot in the 3-sphere never give orientation-preservingly homeomorphic manifolds, the meridian included through Gordon and Luecke. The episode presents the new part as the last case, slopes +2 and −2 on a genus-two knot, states that the reduction to that case is cited rather than reproved and rests partly on unpublished work of Daemi, Lidman and Miller Eismeier, and states that chirally cosmetic surgery, knots in other 3-manifolds, links and Gordon's general conjecture are not claimed. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 19 October 2026: posted as a preprint: OpenAI, 144 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no formalization for this item. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
