Never the same twice?
An AI preprint claims the purely cosmetic surgery conjecture for knots in the 3-sphere
Covers result 306 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:36. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Dehn surgery on a knot in the 3-sphere removes a thickened neighbourhood of the knot and glues a solid torus back, so that a curve of slope p/q bounds the new disc. The first homology of the result is cyclic of order |p|, so slopes whose numerators differ in size always give different manifolds. The question is whether two different slopes on the same nontrivial knot can ever give the same manifold, with orientation. Gordon conjectured, for general manifolds with torus boundary, that they cannot; for knots in the 3-sphere this is the purely cosmetic surgery conjecture. With orientation reversed it does happen: on an amphichiral knot such as the figure-eight, slopes r and −r give mirror images, and on the right-handed trefoil slopes 9 and 9/2 do, found by Mathieu. Gordon and Luecke settled the meridian in 1989. Boyer and Lines, Ni and Wu, and Hanselman cut the possibilities down to slopes ±2 on a genus-two knot, or ±1/q; Daemi, Lidman and Miller Eismeier ruled out the second option; and Futer, Purcell and Schleimer checked more than 350 million prime knots, through 19 crossings.
Result 306 is one OpenAI paper, 144 pages, dated September 23, 2026, and made public October 6. It claims that distinct slopes on a nontrivial knot in the 3-sphere never give orientation-preservingly homeomorphic manifolds, meridian included, through Gordon and Luecke. The new part is the last case: slopes +2 and −2 on a genus-two knot. The reduction to that case is cited, not reproved, and rests partly on work by Daemi, Lidman and Miller Eismeier that is not yet published.
What it doesn’t claim
Chirally cosmetic surgery, knots in other 3-manifolds, links, or Gordon’s more general conjecture for manifolds with torus boundary.
Where it stands
- Posted as a preprint
- OpenAI, 144 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No formalization for this item
Last checked October 19, 2026. The paper has not been revised, and there is no Lean formalization for this item. In the first twelve days we found no public specialist comment and no reported error. OSVerify, an automated referee, reported no major issues and three minor slips, each, it says, leaving the argument intact; its site says experts check its findings but names none, and it is not peer review.
How the proof goes
- A stack of cobordisms. The proof assumes an orientation-preserving homeomorphism between −2 and +2 surgery. It glues surgery cobordisms through that homeomorphism into a stack capped by symplectic pieces, and a count of instantons on the stack gives a nonzero number.
- Ends that must cancel. A moduli space of PU(2) monopoles on the same stack, cut down by phase conditions, is one-dimensional. Its ends are 2^η times that count, at the instantons, plus ends at lens spaces that cancel in pairs. The signed ends of a compact 1-manifold sum to zero, so the count would be zero: a contradiction. Most of the paper’s second half, about 60 pages, rules out every other kind of end.
The numbers in the video
- Ours: on four torus knots, with numerators up to 400, no purely cosmetic pairs and 13 chirally cosmetic pairs on the trefoil, starting with 9/2 and 9.
- Ours: of the 2,977 prime knots up to 12 crossings (KnotInfo data), Boyer and Lines leave 375, Ni and Wu 151, Hanselman 25, Daemi, Lidman and Miller Eismeier’s Alexander polynomial condition 3, and their Jones polynomial condition none. All of these were already known to be fine.
- Ours: the lattice of one piece of the paper’s stack is unimodular, with one positive direction and signature −4.
Read the source
- Purely Cosmetic Surgery on Knots in the Three-Sphere (PDF), 144 pages.
- KnotInfo, the knot table used for our filters.
- The collection’s catalog, where this is result 306.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Dehn surgery on a knot in the 3-sphere removes a thickened neighbourhood of the knot and glues a solid torus back, so that a curve of slope p/q bounds the new disc. Our computer checks that the first homology of the result is cyclic, of order the size of p, for all 185 slopes it tried, so slopes whose numerators differ in size always give different manifolds. The question is whether two different slopes on the same nontrivial knot can ever give the same manifold, with orientation. Gordon conjectured, for general manifolds with torus boundary, that they cannot; Kirby’s problem list records it as the cosmetic surgery conjecture, and for knots in the 3-sphere it is the purely cosmetic surgery conjecture. Orientation matters. Allowing orientation to reverse, it does happen: on an amphichiral knot, such as the figure-eight, slopes r and −r give mirror-image manifolds, and on the right-handed trefoil, slopes 9 and 9/2 give mirror images, found by Mathieu. Our computer searched surgeries on four torus knots, numerators up to 400: no purely cosmetic pairs, but 13 chirally cosmetic pairs on the trefoil, starting with 9/2 and 9. The meridian case was settled in 1989 by Gordon and Luecke: no other slope gives back the 3-sphere. Restrictions then piled up. Boyer and Lines, using an extension of the Casson invariant, then Ni and Wu, and Hanselman, using Heegaard Floer homology, cut the possibilities down to slopes ±2 on a genus-two knot, or ±1/q; Daemi, Lidman and Miller Eismeier then ruled out the second option. Futer, Purcell and Schleimer checked more than 350 million prime knots, through 19 crossings. Our computer runs the filters on the 2,977 prime knots up to 12 crossings: Boyer and Lines leave 375, Ni and Wu 151, Hanselman 25, Daemi, Lidman and Miller Eismeier’s Alexander polynomial condition 3, and their Jones polynomial condition none. All of these were already known to be fine.
2:14The claim
Result 306 is one OpenAI paper, 144 pages, dated September 23 and made public October 6, 2026. It claims that distinct slopes on a nontrivial knot in the 3-sphere never give orientation-preservingly homeomorphic manifolds, meridian included, through Gordon and Luecke. The new part is the last case: slopes +2 and −2 on a genus-two knot. The reduction to that case is cited, not reproved, and rests partly on work by Daemi, Lidman and Miller Eismeier that is not yet published. Not claimed: chirally cosmetic surgery, knots in other 3-manifolds, links, or Gordon’s more general conjecture for manifolds with torus boundary.
3:01How the proof goes
The proof assumes an orientation-preserving homeomorphism between −2 and +2 surgery. It glues surgery cobordisms through that homeomorphism into a stack capped by symplectic pieces, and a count of instantons on the stack gives a nonzero number. A moduli space of PU(2) monopoles on the same stack, cut down by phase conditions, is one-dimensional. Its ends are 2^η times that count, at the instantons, plus ends at lens spaces that cancel in pairs. The signed ends of a compact 1-manifold sum to zero, so the count would be zero: a contradiction. Most of the paper’s second half, about 60 pages, rules out every other kind of end. Our computer checks the lattice of one piece of the stack: unimodular, with one positive direction and signature −4.
3:56Where it stands
The paper is unrefereed and unchanged since release, and there is no Lean formalization. In the first 12 days we found no public specialist comment and no reported error. OSVerify, an automated referee, reported no major issues and three minor slips, each, it says, leaving the argument intact; its site says experts check its findings but names none, and it is not peer review. If it holds, once orientation is kept, surgery on a nontrivial knot in the 3-sphere never repeats itself: different slopes always give different oriented manifolds.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.