CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An abstract annulus need not be a smooth concordance

Statement refuted

Every abstract oriented cobordism between two oriented knot circles in the standard three-sphere can be realized by a smoothly embedded concordance annulus in $S^{3} \times [0, 1]$ .

Statement references

Abstract oriented cobordisms of knot circles

Smooth embedded annulus concordance

Facts & Assumptions

Given: The standard unknot $U$ , the standard right-handed trefoil $T$ , and the claim in the Statement refuted.

[F1]

An abstract oriented cobordism is a compact oriented surface with boundary $- C_{0} ⊔ C_{1}$ and contains no ambient embedding data. (Abstract oriented cobordism of knot circles)

[F2]

The Seifert matrix convention in the library is $V_{i j} = lk (a_{i}^{+}, a_{j})$ for a chosen oriented surface and basis; the surface and basis are part of the data until invariance is proved. (Seifert pairing and matrix of an oriented knot)

[F3]

A trefoil Seifert surface admits a rank-two Seifert matrix whose symmetric sum is $(\begin{array}{cc} - 2 & 1 \\ 1 & - 2 \end{array})$ ; if the cited convention uses the transpose matrix, the symmetric sum is unchanged. For a real symmetric matrix, its signature is the number of positive eigenvalues minus the number of negative eigenvalues. (Collins, Theorem 2.2.9 and Example 2.2.13, https://era.ed.ac.uk/bitstream/handle/1842/5034/Collins2011.pdf?sequence=4)

[F4]

The standard unknot bounds a disk in $S^{3}$ ; the disk is contractible, so its first homology is zero and its symmetrized Seifert form has signature $0$ . (Oriented connected sum and prime knot)

[F5]

A metabolizer for a rank-two integral Seifert form is a rank-one direct summand $L$ with $V (x, y) = 0$ for $x, y \in L$ . Thus every nonzero $x \in L$ satisfies $x^{T} (V + V^{T}) x = 2 V (x, x) = 0$ . This is immediate from the defining bilinear relation, with no signature-invariance or general signature-vanishing theorem. (Metabolic Seifert forms and algebraic knot concordance)

[F6]

Under the Axiom of Countable Choice, a knot is smoothly slice if and only if it is smoothly concordant to the standard unknot. (Smooth sliceness is equivalent to concordance to the unknot)

[A1]

Full Axiom of Choice is assumed by the supplied Seifert-pairing definition. (The Axiom of Choice)

[A2]

Full Axiom of Choice implies the countable-choice hypothesis used by [F6]. (AC implies DC implies countable choice)

[F7]

Every locally flat slice disk, including every smooth slice disk, induces a metabolizer for the positive-push Seifert form of each specified smooth Seifert surface. The proof uses a locally flat hypersurface and smooth chains only in disjoint smooth ambient neighborhoods; it requires no smoothing of that hypersurface. A slice disk induces a Seifert metabolizer, A locally flat slice surface bounds a Seifert hypersurface

Counterexample

technique · direct finite-form obstruction using the slice-to-metabolizer lemma

1.1F1givenconstruct

Let $C_{0}$ be the oriented circle underlying $U$ and $C_{1}$ the oriented circle underlying $T$ . Take the abstract annulus $S^{1} \times [0, 1]$ and identify its lower and upper boundary circles orientation-preservingly with $C_{0}$ and $C_{1}$ , orienting the annulus so its boundary is $- C_{0} ⊔ C_{1}$ . By [F1] this is an abstract oriented cobordism from $U$ to $T$ .

1.2F2F3F4F5A1algebra

For the fixed trefoil matrix in [F3], $V + V^{T} = (\begin{array}{cc} - 2 & 1 \\ 1 & - 2 \end{array}) .$ Its characteristic polynomial is $λ^{2} + 4 λ + 3 = (λ + 1) (λ + 3)$ , so its eigenvalues are $- 1, - 3$ and its signature is $- 2$ . More directly, for $x = (a, b) \neq 0$ over $Q$ or $R$ , $x^{T} (V + V^{T}) x = - 2 (a^{2} - a b + b^{2}) = - (a - b)^{2} - a^{2} - b^{2} < 0.$ Consequently $V$ cannot have a rank-one metabolizer: a nonzero vector in one would have zero value by [F5]. This is a calculation for the displayed Seifert surface and basis; no knot-level signature invariance is being inferred. The unknot's rank-zero form in [F4] still has signature zero.

2.1A2F3F5F6F7step 1.1step 1.2∎

If $U$ were smoothly concordant to $T$ , reverse the interval coordinate to obtain smooth concordance of $T$ to the unknot. By [F6] and [A2], $T$ would bound a smooth proper disk. Apply [F7] to the particular trefoil Seifert surface in [F3]. It gives a rank-one metabolizer for $V$ , contradicting the strict negative-definiteness calculation and [F5] in step 1.2. Thus the displayed abstract oriented annulus has no realization as a smooth concordance between its specified endpoint knots, refuting the claimed universal realization. The locally flat slice-to-metabolizer lemma applies because every smooth proper disk is locally flat.

Sources

Abstract oriented cobordism of knot circles

Smooth knot concordance

Smooth sliceness is equivalent to concordance to the unknot

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

44 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

Charles Livingston, A Survey of Classical Knot Concordance, Handbook of Knot Theory, Chapter 7 (standard reference, not scraped)
Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)
Julia Collins, On the concordance orders of knots, PhD thesis (standard reference, not scraped)