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

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An invertible Seifert matrix gives a rational isometric structure

Statement

Assume full AC for the knot Seifert-pairing convention. Let $V \in M_{2 g} (Z)$ be an invertible Seifert matrix and put $B = V + V^{T}$ and $T = V^{- 1} V^{T}$ on $Q^{2 g}$ . Then $B$ is a nonsingular symmetric bilinear form, $T$ is an invertible $B$ -isometry, and $I - T$ is invertible; hence $(Q^{2 g}, B, T)$ is a rational Seifert isometric structure.

Statement references

The positive-push integral Seifert matrix

The definition of a rational Seifert isometric structure

Facts & Assumptions

Given: Full AC and an invertible integral Seifert matrix $V$ .

[A1]

Full AC is the stated hypothesis of the Seifert-pairing interface. The Axiom of Choice

[F1]

For a knot Seifert matrix, $V - V^{T}$ is the unimodular surface intersection matrix. The skew part of a Seifert matrix is the surface intersection form

[F2]

A rational Seifert isometric structure has a nonsingular symmetric form $B$ , an invertible $B$ -isometry $T$ , and invertible $I - T$ . Integral Witt classes and rational Seifert isometric structures

Proof

1.1A1F1givenalgebra

The matrix $B = V + V^{T}$ is symmetric. Modulo $2$ , $V + V^{T} \equiv V - V^{T}$ ; by [F1] the determinant of $V - V^{T}$ is odd, so $\det B$ is odd and in particular nonzero. Thus $B$ is nonsingular over $Q$ .

2.1F2step 1.1algebra

Since $V$ and $V^{T}$ are invertible over $Q$ , $T = V^{- 1} V^{T}$ is invertible. Direct multiplication gives $T^{T} B T = V (V^{T})^{- 1} (V + V^{T}) V^{- 1} V^{T} = V + V^{T} = B,$ because the two summands in the middle expression simplify respectively to $V$ and $V^{T}$ . Therefore $T$ is a $B$ -isometry.

3.1F1F2step 2.1algebra∎

Finally $I - T = V^{- 1} (V - V^{T})$ , which is invertible by invertibility of $V$ and [F1]. The three defining conditions in [F2] hold, proving the claim.

Depends on

Used by

Unimodular skew part does not force metabolicity Counterexample
Worked cases for metabolic Seifert forms Example

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Sources

Charles Livingston, The concordance genus of a knot, II (standard reference, not scraped)
Charles Livingston, A Survey of Classical Knot Concordance (standard reference, not scraped)