Browse › Topology › 306 The purely cosmetic surgery conjecture

Purely cosmetic surgery on knots in the three-sphere

OpenAI · September 23, 2026

Open PDF Cite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We prove the purely cosmetic surgery conjecture for knots in the three-sphere. Distinct Dehn surgery slopes on a nontrivial smooth knot never give orientation-preservingly homeomorphic manifolds.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

Kazuhiro Ichihara, Purely cosmetic surgeries on knots with low-span Jones polynomials (arXiv:2609.32507, 26 September 2026). Independently proves that knots whose nontrivial Jones polynomial has span at most 11 admit no purely cosmetic surgery, a partial result on the case left after the manuscript's Proposition 1.2 reduction, which uses the Jones-polynomial conditions of Daemi–Lidman–Miller Eismeier.
Kazuhiro Ichihara and Kengo Kishimoto, Purely cosmetic surgeries on knots with low-span Jones polynomials, II (arXiv:2610.10019, 7 October 2026). Extends the Jones-span exclusion to span at most 14 and covers odd-span knots of braid index at most 4, an independent partial result on the conjecture claimed in full by Theorem 1.1; its arXiv page does not mention the OpenAI release.
Kazuhiro Ichihara, In Dae Jong and Yasuyoshi Tsutsumi, Purely cosmetic surgeries and Casson–Walker–Lescop invariants (arXiv:2603.11720, 12 March 2026). Uses a rational surgery formula for the Casson–Walker–Lescop invariant to constrain purely cosmetic surgeries, continuing the classical-invariant obstructions (Boyer–Lines) surveyed in the manuscript's introduction.
F. M. Brady, Using the Jones Polynomial to Prove Infinite Families of Knots Satisfy the Cosmetic Surgery Conjecture (arXiv:2512.20511, December 2025, revised January 2026). Verifies the conjecture for two infinite knot families using Jones-polynomial obstructions, in the line of the Ichihara–Wu obstructions the manuscript cites among earlier special cases.

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 306

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026,
  author = {{OpenAI}},
  title = {{Purely cosmetic surgery on knots in the three-sphere}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf}{OAI:Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026}},
  year = {2026}
}