LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

The square Seifert presentation generates the knot Alexander polynomial

Statement

Assume the full Axiom of Choice. Let $K \subset S^{3}$ be a smooth oriented knot, let $F \subset S^{3}$ be a connected compact smooth oriented embedded surface with oriented boundary $K$ , and let $χ (F) = 1 - 2 g$ with $g \geq 0$ . Choose an ordered integral basis $a_{1}, \dots, a_{2 g}$ of $H_{1} (F; Z)$ , use the positive normal convention, and set $V_{i j} = lk (a_{i}^{+}, a_{j})$ . Let $Λ = Z [t^{\pm 1}]$ , let $A_{K}$ be the first homology of the infinite cyclic cover classified by total linking with $t$ acting by the positive-meridian deck transformation, and let $Δ_{K}$ be a generator of the zeroth elementary ideal $E_{0} (A_{K})$ . Then the square matrix $M (t) = V - t V^{T}$ generates that ideal:

$E_{0} (A_{K}) = (\det M (t)) .$

For $g = 0$ , the matrix is empty and its determinant is $1$ . In every genus, $\det (V - V^{T}) = \pm 1$ , so every generator $Δ_{K}$ satisfies $Δ_{K} (1) = \pm 1$ . The ideal and determinant assertions are compatible with integral basis changes and with orientation-preserving ambient isotopies of knot-surface pairs.

Statement references

The definition and genus convention for a Seifert surface

The positive-push Seifert matrix convention

The one-variable Alexander module and positive deck action

The Alexander polynomial as a generator of the zeroth elementary ideal

Facts & Assumptions

Given: Full AC, an oriented knot $K \subset S^{3}$ , a connected compact oriented Seifert surface $F$ with $\partial F = K$ , and an ordered integral basis of $H_{1} (F; Z)$ .

[A1]

Full AC is the assertion that every family of nonempty sets has a choice function. It is the stated hypothesis of the Alexander-module and Seifert-matrix presentation conventions used here. (The Axiom of Choice)

[F8]

Full AC implies DC and hence the Axiom of Countable Choice, which is the stated choice hypothesis of the Seifert-surface and finite surface-normal-form suppliers. (AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

A Seifert surface is connected, compact, smooth, oriented and embedded with the given knot as oriented boundary; in the chosen basis the positive-push matrix has entries $V_{i j} = lk (a_{i}^{+}, a_{j})$ . (Oriented Seifert surfaces in the three-sphere, Seifert pairing and matrix of an oriented knot)

[F2]

The one-variable Alexander module is defined over $Λ = Z [t^{\pm 1}]$ ; $Δ_{K}$ is a greatest common divisor of $E_{0} (A_{K})$ and is determined up to a unit $\pm t^{k}$ . (The one-variable Alexander module of an oriented link, The Alexander polynomial from the zeroth elementary ideal, Elementary ideals of a finitely presented module, The Laurent polynomial ring as the principal localisation of Z[t] at t, Units, powers and the domain property of the Laurent polynomial ring)

[F3]

The elementary ideal $E_{0}$ is independent of the finite presentation and depends only on the module's isomorphism class. (Elementary ideals are independent of the presentation)

[F4]

The positive-meridian convention gives an isomorphism $A_{K} ≅ coker (V - t V^{T})$ over $Λ$ . (A Seifert matrix presents the one-variable Alexander module)

[F5]

The skew part of the matrix is the surface intersection matrix: $V - V^{T} = J_{F}$ , and $J_{F}$ is unimodular. (The skew part of a Seifert matrix is the surface intersection form)

[F6]

Under an orientation-preserving ambient isotopy, a transported basis has the same Seifert matrix; a target basis change gives $V^{'} = P^{T} V P$ for $P \in {G L}_{2 g} (Z)$ . (Seifert matrices of ambient-isotopic pairs are S-equivalent)

[F7]

An orientation-preserving ambient isotopy induces an isomorphism of Alexander modules and preserves $E_{0} (A_{K})$ and $Δ_{K}$ up to a unit. (The Alexander polynomial is an oriented link invariant)

Proof

1.1A1F1F8F2F3F4givenalgebra

Put $n = 2 g$ and $M (t) = V - t V^{T}$ . By [F1] this is the positive-push matrix in the Statement, and by [F4] $A_{K}$ is isomorphic to the cokernel of the square presentation matrix $M (t)$ . The definition of $E_{0}$ uses the $n \times n$ minors; the only such minor of an $n \times n$ matrix is its determinant. Thus the presentation gives $E_{0} (coker M) = (\det M)$ , including $n = 0$ , where the sole empty minor is $1$ . Presentation and isomorphism independence [F3] yields $E_{0} (A_{K}) = (\det M)$ .

2.1F2F5step 1.1algebra

Evaluating at $t = 1$ gives $\det M (1) = \det (V - V^{T}) = \det J_{F} = \pm 1$ by [F5]. Since $E_{0} (A_{K}) = (\det M)$ and $Δ_{K}$ is a generator of that principal ideal, $Δ_{K} = u \det M$ for a unit $u = \pm t^{k}$ by [F2]; every such unit evaluates to $\pm 1$ at $t = 1$ . Hence $Δ_{K} (1) = \pm 1$ .

3.1F6F7step 1.1givenalgebra∎

If the basis is changed by $P \in {G L}_{n} (Z)$ , then $M^{'} (t) = P^{T} M (t) P$ and $\det M^{'} = (\det P)^{2} \det M = \det M$ . Under an ambient isotopy, [F6] gives this same transported-matrix relation and [F7] preserves the Alexander ideal up to its unit generator. Thus the asserted ideal-generator relation is compatible with the stated choices.

Depends on

Used by

Knot Alexander normalization, symmetry, and genus bound Corollary
The knot Alexander polynomial multiplies under connected sum Corollary
A higher cyclic branched cover can have infinite first homology Counterexample
The knot-normalized Alexander fraction need not be a Laurent polynomial Counterexample
Trefoil and figure-eight Alexander polynomials from Seifert matrices Example
One-sided Levine–Tristram signature limits exist Lemma
The singular parameters of the Levine–Tristram form Lemma
Knot determinant is the order of double-cover homology Theorem
The Seifert determinant is the knot Alexander polynomial Theorem

Dependency tree · two levels

91 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

Dale Rolfsen, Knots and Links (standard reference, not scraped)
Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes (standard reference, not scraped)