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

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Locally flat sliceness implies Fox–Milnor factorization

Statement

Assume the full Axiom of Choice. Let $K \subset S^{3}$ be a smooth oriented knot, let $F \subset S^{3}$ be a connected compact smooth oriented Seifert surface of genus $g$ , and let $V$ be its positive-push Seifert matrix, with entries $V_{i j} = lk (a_{i}^{+}, a_{j})$ in an ordered integral basis of $H_{1} (F; Z)$ . Let $Δ_{K} (t)$ be the Alexander polynomial normalized by $Δ_{K} (t) ≐ \det (V - t V^{T})$ , where $p ≐ q$ means $p = \pm t^{k} q$ for some $k \in Z$ . If the integral Seifert lattice is metabolic, then there is an $f (t) \in Z [t]$ such that $Δ_{K} (t) ≐ f (t) f (t^{- 1})$ . The same factorization is necessary if $K$ bounds a locally flat properly embedded disk in the standard four-ball $B^{4}$ . This is a necessary condition for either algebraic or locally flat sliceness, not a sufficient condition. The factorization concerns the Alexander polynomial $Δ_{K}$ , not the separate knot-normalized fraction $D_{K} (t) = Δ_{K} (t) / (1 - t)$ .

Statement references

The positive-push Seifert matrix convention

The integral metabolizer condition

The determinant normalization of the Alexander polynomial

Facts & Assumptions

Given: Full AC, an oriented knot $K$ , a connected oriented Seifert surface $F$ , its positive-push matrix $V$ , and either a metabolizer for $V$ or a locally flat proper slice disk for $K$ .

[A1]

Full AC is the family-of-nonempty-sets choice axiom used by the knot Seifert-pairing convention. The Axiom of Choice

[F1]

The positive-push matrix represents the integral Seifert pairing, with the stated linking convention. Seifert pairing and matrix of an oriented knot

[F2]

The generator of the zeroth Alexander elementary ideal satisfies $Δ_{K} (t) ≐ \det (V - t V^{T})$ in this convention. The Seifert determinant is the knot Alexander polynomial

[F3]

A metabolic Seifert lattice has a rank- $g$ direct summand on which the pairing vanishes. Metabolic Seifert forms and algebraic knot concordance

[F4]

For any integral rank- $2 g$ bilinear form with unimodular skew part and a rank- $g$ direct-summand metabolizer, its determinant is $t^{g} f (t) f (t^{- 1})$ for some $f \in Z [t]$ . A metabolic Seifert matrix has Fox–Milnor factorization

[F5]

Under full AC the actual slice-metabolizer supplier proves that every locally flat proper slice disk induces the required integral rank- $g$ primitive metabolizer. Its TOP branch uses the explicitly sourced geometric inputs of the Seifert-hypersurface supplier and retains the local duality, isotropy and integral-saturation argument. A slice disk induces a Seifert metabolizer

Proof

1.1A1F1F2F3F4givenalgebra

If the Seifert lattice is metabolic, [F3] supplies a rank- $g$ direct-summand metabolizer for the positive-push matrix [F1]. Its skew part is unimodular by the knot Seifert convention, so [F4] gives $\det (V - t V^{T}) = t^{g} f (t) f (t^{- 1})$ . By [F2], $Δ_{K} (t)$ differs from this determinant by a Laurent unit, proving the algebraic factorization claim. This matrix calculation uses no choice beyond the stated full-AC convention for the Seifert data.

2.1A1F5step 1.1∎

If $K$ bounds the locally flat proper disk of the Statement, [F5] provides the actual integral metabolizer for the positive-push Seifert lattice on the given $F$ . Apply step1.1 to obtain $Δ_{K} (t) ≐ f (t) f (t^{- 1})$ . This includes a smooth slice disk as a special case. The conclusion is the stated necessary condition for $Δ_{K}$ , and does not assert sufficiency or a factorization of $D_{K}$ .

Depends on

Used by

Failure of Fox–Milnor factorization obstructs local sliceness Corollary
The trefoil fails both algebraic sliceness tests Counterexample

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Sources

Ralph H. Fox and John W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots (standard reference, not scraped)
Charles Livingston, A Survey of Classical Knot Concordance (standard reference, not scraped)