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

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The negative Seifert form represents the algebraic inverse

Statement

Assume full AC when the form is obtained from a knot. For every integral Seifert lattice $(H, V)$ , the class represented by $(H, - V)$ is the additive inverse of the class represented by $(H, V)$ in $G_{Z}$ .

Statement references

The group presentation of algebraic concordance and the definition of a metabolic lattice

Facts & Assumptions

Given: An integral Seifert lattice $(H, V)$ and the defining relations of $G_{Z}$ .

[A1]

Full AC is inherited only when the abstract lattice is instantiated as a knot's Seifert pairing; the diagonal calculation below is choice-free. The Axiom of Choice

[F1]

In $G_{Z}$ , direct sum is addition and every metabolic lattice represents zero. Metabolic Seifert forms and algebraic knot concordance

Proof

1.1A1F1givenalgebra

In $(H, V) \oplus (H, - V)$ , the diagonal subgroup $Δ H = {(x, x) : x \in H}$ is a direct summand: it is the kernel of the surjection $(x, y) \mapsto x - y$ and that surjection has section $z \mapsto (z, 0)$ . Its rank is $rank H$ , half the rank of $H \oplus H$ . For $x, y \in H$ , $(V \oplus (- V)) ((x, x), (y, y)) = V (x, y) - V (x, y) = 0$ , so $Δ H$ is a metabolizer.

2.1F1step 1.1algebra∎

By [F1], the metabolic direct sum in step 1.1 represents zero, while the direct-sum relation identifies its class with $[V] + [- V]$ . Hence $[- V] = - [V]$ .

Depends on

Used by

Dependency tree · two levels

9 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 (standard reference, not scraped)