∑math20

Topology · Result 306

The purely cosmetic surgery conjecture

1 paper

Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.

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The problem

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Papers 1

First page of Purely cosmetic surgery on knots in the three-sphere

Purely cosmetic surgery on knots in the three-sphere

OpenAI · September 23, 2026

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.

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@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}
}

Formal verification

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