TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Piccirillo’s partner has the same zero-trace and Rasmussen invariant two

Statement

Assume the Axiom of Choice. For the Conway knot $C$ and Piccirillo partner $K_{P}$ fixed above, the zero-traces are diffeomorphic and $X_{0} (C) ≅ X_{0} (K_{P}), s (K_{P}) = 2.$ This statement concerns the zero-trace $4$ -manifolds; it asserts no smooth concordance between $C$ and $K_{P}$ .

Statement references

Piccirillo, Proposition 1.4 and Theorem 1.5

Hom, §§2–3

Facts & Assumptions

Given: Full AC and the specified knots $C$ and $K_{P}$ .

[A1]

The Axiom of Choice, written AC, is assumed. The Axiom of Choice

[A2]

AC implies dependent choice and hence countable choice ${A C}_{ω}$ ; this supplies every retained local countable-choice hypothesis. AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Piccirillo’s Conway diagrams give a dualizable handle link constructs under full AC one actual framed handlebody with zero-trace descriptions $X_{0} (C)$ and $X_{0} (K_{P})$ . The local witness is identified with $X_{0} (C)$ by its framed construction; its identification with $X_{0} (K_{P})$ composes this with the explicitly imported Proposition 1.4. The retained local endpoint is not asserted to be the published panel $N$ .

[F2]

Lee computations for the Conway knot and its Piccirillo partner supplies the established exact rational Lee calculation on the two specified diagrams, giving $s (C) = 0$ and $s (K_{P}) = 2$ .

Proof

technique · direct

1.1A1F1construct

By [F1], the two zero-traces are diffeomorphic to the same framed handlebody. Compose the first diffeomorphism with the inverse of the second to obtain $X_{0} (C) ≅ X_{0} (K_{P})$ . The supplier uses the published trace equality for the named Figure 2 knot independently of its local endpoint encoding.

2.1A1A2F2step 1.1construct∎

Apply [F2] to that specified diagram to obtain $s (K_{P}) = 2$ . Together with step 1.1 this gives both conclusions of the Statement.

Depends on

Used by

Equal zero-traces do not force smooth concordance Counterexample
The Conway knot is topologically slice but not smoothly slice Theorem

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lisa Piccirillo, The Conway knot is not slice (standard reference, not scraped)
Jennifer Hom, Getting a handle on the Conway knot (standard reference, not scraped)