TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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The Arf invariant is the second Conway coefficient modulo two

Statement

Assume full AC and let $K$ be a smooth oriented knot in the standard oriented $3$ -sphere. With the normalized Conway polynomial $\nabla_{K} (z) = 1 + a_{2} (K) z^{2} + a_{4} (K) z^{4} + \dots$ and the Seifert-form definition of $Arf (K)$ , $Arf (K) = a_{2} (K) (m o d 2) .$

Statement references

Surface independence and the Conway–Arf identity

The normalized Conway convention

Facts & Assumptions

Given: Full AC and a smooth oriented knot $K$ .

[A1]

Full AC is the ambient hypothesis of the Seifert and Alexander conventions. The Axiom of Choice

[F1]

The Seifert Arf invariant and normalized Conway coefficient use the conventions stated in their definitions. Knot Arf invariant from the Seifert form, S-equivalence and Conway normalization

[F2]

The bridge theorem proves the claimed congruence and surface independence. Seifert S-equivalence and the Conway–Arf bridge

Proof

1.1A1F1F2given

Apply [F2] to the knot $K$ with the definitions in [F1]. It gives exactly the displayed equality in $F_{2}$ ; equality of two elements of $F_{2}$ is the stated congruence.

2.1F1F2step 1.1∎

The bridge theorem already proves independence of the chosen surface and basis, so this corollary adds no new representative choice or normalization.

Depends on

Used by

Trefoil and figure-eight Levine–Tristram and Arf calculations Example

Dependency tree · two levels

17 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

W. B. Raymond Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics 175, Springer (1997) (standard reference, not scraped)