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Math-3025 · mathematics · Topology

Pure cosmetic surgery for knots in the three-sphere

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If K is a nontrivial smooth knot in $S^{3}$ and Dehn fillings along slopes r and s are orientation-preservingly homeomorphic, must $r=s$?

Short English statement

If K is a nontrivial smooth knot in $S^{3}$ and Dehn fillings along slopes r and s are orientation-preservingly homeomorphic, must $r=s$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3025-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3025

Manuscript 1 (family 306, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Pure cosmetic surgery for knots in the three-sphere

OpenAI's claim

For any nontrivial smooth knot in $S^3$, the paper claims that surgeries at different slopes cannot be orientation-preservingly homeomorphic, resolving purely cosmetic surgery. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For any smooth nontrivial knot $K\subset S^3$ and slopes $r,s\in\mathbb Q\cup\{\infty\}$, OpenAI claims \[ S^3_r(K)\cong S^3_s(K)\text{ by an orientation-preserving homeomorphism}\ \Longrightarrow\ r=s. \] Slopes use the meridian and preferred longitude; $p/q$ represents the primitive class $p\mu+q\lambda$ up to simultaneous sign. The orientation of the Dehn filling extends the knot exterior's orientation. This assertion concerns purely cosmetic surgery on a single knot in $S^3$, excluding the unknot and orientation-reversing equivalences.

Outline of the argument

Manuscript 1 (family 306, 2026-09-23): Existing Floer results confine a possible cosmetic pair to slopes +2 and -2 with knot genus two. Integral determinant lifts produce a nonzero instanton count, while toric period and tube estimates exclude unwanted reducible and broken strata. Compatible spinor regularization then builds an oriented one-dimensional count whose allowed boundary would force a nonzero multiple of that instanton count to vanish.

Formal verification

Family 306 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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