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

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Failure of Fox–Milnor factorization obstructs local sliceness

Statement

Assume the full Axiom of Choice. Let $K \subset S^{3}$ be a smooth oriented knot and let $Δ_{K} (t)$ be its Alexander polynomial in the determinant normalization, with $p ≐ q$ meaning $p = \pm t^{k} q$ for some $k \in Z$ . If there is no Laurent polynomial $f (t) \in Z [t^{\pm 1}]$ such that $Δ_{K} (t) ≐ f (t) f (t^{- 1})$ , then $K$ does not bound a locally flat properly embedded disk in the standard four-ball $B^{4}$ . This is a necessary obstruction only; factorization alone does not assert sliceness.

Statement references

The necessary factorization for a locally flat slice disk

Facts & Assumptions

Given: Full AC, an oriented knot $K$ , and the hypothesis that no Laurent polynomial $f (t) \in Z [t^{\pm 1}]$ gives the stated reciprocal-norm factorization of $Δ_{K} (t)$ .

[A1]

Full AC is the choice axiom assumed by the determinant-normalized knot Seifert-pairing convention. The Axiom of Choice

[F1]

A locally flat slice disk implies the stated integral reciprocal-norm factorization, by the full TOP Fox–Milnor theorem with its explicit geometric inputs. Locally flat sliceness implies Fox–Milnor factorization

Proof

1.1A1F1given

Suppose, to the contrary, that $K$ bounds a locally flat proper disk in $B^{4}$ . The full TOP implication [F1] gives a Laurent polynomial $f (t)$ with $Δ_{K} (t) ≐ f (t) f (t^{- 1})$ . The full-AC hypothesis is the same as that of the supplier.

2.1givenstep 1.1∎

This contradicts the assumed nonexistence of such an $f$ . Therefore no such locally flat slice disk exists. Only the necessary implication was used; nothing asserts that reciprocal-norm factorization is sufficient for sliceness.

Depends on

Used by

The trefoil fails both algebraic sliceness tests Counterexample

Dependency tree · two levels

8 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

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)