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

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A certified knot-spine Wada quotient equals twisted Reidemeister torsion

Statement

Assume the Axiom of Choice convention of the twisted-chain definition. Let $K \subset S^{3}$ be an oriented knot, let $E_{K}$ be its compact exterior, let $X$ be a finite based deficiency-one two-dimensional spine with one vertex, $n \geq 2$ oriented $1$ -cells and $n - 1$ oriented $2$ -cells, certified by a based cellular simple-homotopy equivalence with $E_{K}$ , and let $ψ : π_{1} (E_{K}) ↠ H ≅ Z^{m}$ have $m \geq 1$ . Let $ρ : π_{1} (E_{K}) \to {G L}_{d} (F)$ be a representation over a field, and assume the degree-one twisted homology over $Λ = F [H]$ is a torsion module. For any admissible generator column of the Fox block matrix of the certified spine, its deleted-minor Wada quotient represents the based twisted Reidemeister torsion of $E_{K}$ in $Q^{\times} / U_{ρ, ψ}$ , where $Q = Frac (Λ)$ and $U_{ρ, ψ} = {\pm \det (ρ (g)) ψ (g)^{d} : g \in π_{1} (E_{K})}$ . All admissible columns give the same class. If two such deficiency-one spines are related by a supplied finite based simple cellular equivalence carrying the same representation and deck map, their Wada quotient classes also agree. A group-presentation isomorphism alone is not such an equivalence.

Statement references

Admissible deleted-column quotient

Fox matrix as cellular boundary

Exterior-spine acyclicity

Deleted-minor torsion identity

Simple-spine unit ambiguity

Kirk and Livingston, §§3–4

Conway, §§2.3–2.4

Ranicki, basis changes and Theorem 15

Facts & Assumptions

Given: AC; the finite based knot-exterior spine certificate; a positive-rank free-abelian deck epimorphism; a field-valued representation; degree-one Laurent-ring homology torsion; and an admissible deleted column.

[A1]

AC is inherited from the maximal-abelian-cover convention. The Axiom of Choice; Twisted chain complex and based Reidemeister torsion for link exteriors

[F1]

The evaluated Fox block matrix of a based presentation complex is the cellular boundary matrix of its twisted chain complex, with the stated row-coefficient convention. The evaluated Fox Jacobian is the twisted cellular boundary

[F2]

Torsion degree-one homology, positive-rank deck map, and the stated deficiency-one cell counts imply acyclicity over the fraction field. Torsion first homology makes the localized exterior spine complex acyclic

[F3]

For an admissible deleted column of a certified deficiency-one spine, the Fox minor divided by its boundary determinant is the based twisted torsion class, and all admissible columns agree. The admissible Wada deleted minor equals based twisted torsion

[F4]

Changing lifts, orientations or the certified simple cellular model changes torsion only by an element of the determinant and deck-unit subgroup. Based twisted torsion changes only by determinant and deck units

[F5]

The twisted-chain definition gives the based fraction-field torsion and its exact unit quotient. Twisted chain complex and based Reidemeister torsion for link exteriors

Proof

1.1givenF1

The supplied simple-spine certificate identifies the cellular presentation complex with the selected knot exterior model and transports the based cells, representation, and deck map. Therefore [F1] identifies the evaluated Fox matrix with the actual twisted cellular boundary matrix of this model; no conclusion here follows merely from an isomorphism of presented groups.

1.2A1F2F5given

The spine has one $0$ -cell, $n$ $1$ -cells and $n - 1$ $2$ -cells, and the deck map has positive rank. The given torsion hypothesis is exactly the degree-one module hypothesis in [F2]. Hence its fraction-field complex is acyclic, so the torsion class in [F5] is defined and nonzero.

2.1F1F3step 1.2

The chosen column is admissible by hypothesis. Apply [F3] to the based spine and the acyclic complex from step 1.2. It states that the deleted Fox minor divided by the selected $d_{1}$ block determinant represents the torsion class in $Q^{\times} / U_{ρ, ψ}$ , and it proves the same conclusion for every other admissible column.

3.1F3F4F5step 2.1algebra∎

If a second certified deficiency-one spine is related to the first by the stated finite based simple cellular equivalence, [F4] makes its based torsion class equal in the same quotient; the representation and deck map are transported by that equivalence, so the subgroup $U_{ρ, ψ}$ is unchanged. Applying step 2.1 to the second spine shows its admissible Wada quotient has that same class. This proves the permitted presentation-change assertion without promoting a group isomorphism to a spine equivalence.

Depends on

Used by

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Sources

Paul Kirk and Charles Livingston, Twisted Alexander Invariants, Reidemeister Torsion, and Casson–Gordon Invariants, §§3–4 (standard reference, not scraped)
Anthony Conway, Burau Maps and Twisted Alexander Polynomials, §§2.3–2.4 (standard reference, not scraped)
Andrew Ranicki, Notes on Reidemeister Torsion, basis-change discussion and Theorem 15 (standard reference, not scraped)