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Math-2204 · mathematics · Topology

Kirby Problem 1.12

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a knot in an oriented three-manifold, Dehn filling along a slope on its peripheral torus produces a closed three-manifold. Two fillings are purely cosmetic if their resulting manifolds are orientation-preservingly homeomorphic; they are chirally cosmetic if an orientation-reversing homeomorphism identifies the results. The question has four parts. Can inequivalent slopes ever be purely cosmetic? If two knots in a closed oriented three-manifold have orientation-preservingly homeomorphic complements, must an ambient homeomorphism carry one knot to the other? Does every nontrivial knot in $S^3$ admit no purely cosmetic pair? Finally, must a knot in $S^3$ other than a torus knot $T_{2g+1,2}$ admit no chirally cosmetic pair of inequivalent slopes?

Formulation references: [S1], [S2], [S3], [S4].

Short English statement

For a knot in an oriented three-manifold, Dehn filling along a slope on its peripheral torus produces a closed three-manifold. Two fillings are purely cosmetic if their resulting manifolds are orientation-preservingly homeomorphic; they are chirally cosmetic if an orientation-reversing homeomorphism identifies the results. The question has four parts. Can inequivalent slopes ever be purely cosmetic? If two knots in a closed oriented three-manifold have orientation-preservingly homeomorphic complements, must an ambient homeomorphism carry one knot to the other? Does every nontrivial knot in $S^3$ admit no purely cosmetic pair? Finally, must a knot in $S^3$ other than a torus knot $T_{2g+1,2}$ admit no chirally cosmetic pair of inequivalent slopes?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2204-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2204

Manuscript 1 (family 306, 2026-09-23): Theorem 1.1 proves clause (c) for every nontrivial S3 knot, but the bundle also asks cosmetic/complement questions in arbitrary closed three-manifolds and a chirally cosmetic classification, so the bundle is not wholly resolved.

Kirby Problem 1.12

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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