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CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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The trefoil fails both algebraic sliceness tests

Statement refuted

Assume full AC, and let T be the right-handed trefoil whose positive-push Seifert matrix in the ordered Hopf-core basis is V=(−10−1−1). The claim refuted is that T bounds a locally flat properly embedded disk in the standard four-ball B4.

Facts & Assumptions

Given: Full AC, the positive-push trefoil matrix V, and the claim that the trefoil is locally flat slice to be refuted.

[A1]

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

[F1]

The positive Hopf-plumbed right-handed trefoil has V=(−10−1−1) in the stated basis. A positive Hopf plumbing gives the trefoil Seifert matrix

[F2]

The determinant det⁡(V−tVT) generates the Alexander polynomial up to a Laurent unit. The Seifert determinant is the knot Alexander polynomial

[F3]

At a nonroot ω≠1, the Levine–Tristram signature is the signature of Hω(V)=(1−ω)V+(1−ωˉ)VT. Levine–Tristram Hermitian form and signature

[F4]

If this Seifert form is metabolic and ΔT(ω)≠0, then σω(T)=0. A metabolizer forces Levine–Tristram signatures to vanish

[F5]

A locally flat slice disk gives a primitive Seifert metabolizer and hence the Fox–Milnor reciprocal norm, with the full-AC geometric and algebraic inputs explicitly stated in those suppliers. Locally flat sliceness implies Fox–Milnor factorization, A slice disk induces a Seifert metabolizer

[F6]

Failure of this reciprocal-norm condition obstructs a locally flat slice disk, by the contrapositive corollary. Failure of Fox–Milnor factorization obstructs local sliceness

Counterexample

1.1A1F1F2givenalgebra

By [F1] and [F2], ΔT(t)≐det⁡(V−tVT)=(1−t)2+t=t2−t+1. Thus ∣ΔT(−1)∣=3, since every Laurent unit ±tk evaluates to ±1 at −1.

2.1step 1.1algebra

If ΔT(t)≐f(t)f(t−1) for some f∈Z[t±1], then evaluating at t=−1 gives 3=f(−1)2 or −3=f(−1)2. Neither is possible because an integer square is nonnegative and 3 is not an integer square. Hence the trefoil polynomial is not an integral reciprocal norm.

2.2F1F3F4step 1.1algebra

At ω=−1, one has H−1(V)=2(V+VT)=(−4−2−2−4), whose eigenvalues are −2 and −6. Since ΔT(−1)≠0, its Levine–Tristram signature is −2. By [F4], a metabolic Seifert form would have signature 0 there, so the trefoil Seifert form is not metabolic.

3.1F4F5F6step 2.1step 2.2∎

By [F6] and step2.1, the trefoil does not bound a locally flat properly embedded disk in the standard four-ball. Independently, if such a disk existed then [F5] would give a metabolic Seifert form, contradicting the signature calculation in step2.2 and [F4]. Thus the original locally flat sliceness claim is refuted by both necessary tests, with their TOP geometric inputs supplied rather than inferred from the numbers alone.

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