Topology·OpenAI Math·Result 306
The purely cosmetic surgery conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under topology: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of the purely cosmetic surgery conjecture (topology) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.
Chapter 13 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 4:12 · watch from there on YouTube →
What OpenAI claims
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.
Paper
Purely cosmetic surgery on knots in the three-sphere
OpenAI's 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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.
- Where is the proof?
- In the paper, Purely cosmetic surgery on knots in the three-sphere (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 13).
Also in
- Chapter 2 of 10 in Kervaire Invariant Problem at Prime 3, The Grothendieck Homotopy Hypothesis and 8 More · starts at 0:21 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · Compilation · 29:20 · chapter 13 from 4:12 · description, sources and chapters →
Related results in Topology
- 307Failure of rational injectivity for maximal coarse assemblyVideoLean
- 308Finite Smith–Toda complexes at every heightVideo
- 309The Kervaire invariant problem at the prime threeVideo
- 310Quillen's conjecture in rational homologyVideo
- 311The Hovey–Strickland and Chai conjecturesVideo
- 312The Grothendieck homotopy hypothesisVideoLean
