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The colored twisted Burau procedure computes exterior torsion

Statement

Assume the Axiom of Choice convention of the twisted-chain definition. Let a finite colored geometric braid, a field with exact arithmetic, a representation of the closure exterior group, the full positive-meridian abelianization followed by a declared surjective color map, and compatible based spine and axis-filling certificates satisfy all input conditions of the colored twisted Burau procedure. Assume its designated denominator $D = \det (ρ (g_{n}) ψ (g_{n}) - I_{d})$ is nonzero, so the procedure is applicable, and assume the representation and deck map extend over the filling with the axis meridian acting trivially. Then the procedure returns the class of the twisted Reidemeister torsion of the closure exterior. If the fraction-field complex is acyclic, the returned value is $N / D$ , where $N = \det ({\overline{B}}_{ρ} (β) - I_{(n - 1) d})$ ; if it is nonacyclic, the procedure returns $0$ and $N = 0$ , in agreement with the declared zero-torsion convention. The geometric certificates are essential hypotheses: matching a group presentation alone does not certify an exterior spine or the axis-filling decomposition.

Statement references

Finite colored twisted Burau procedure and its input boundary

Braid-axis torsion product

Exterior-spine Wada quotient theorem

Twisted coefficient and torsion convention

Conway, Theorem 3.15

Kirk and Livingston, §§2–4

Ranicki, Definitions 4 and 6 and Theorem 15

Facts & Assumptions

Given: AC; all finite encoded procedure inputs; a nonzero designated denominator; the actual based spine and compatible axis-filling certificates; and the stated coefficient extension across filling.

[A1]

AC is inherited from the full abelianization and twisted-chain convention. The Axiom of Choice; Twisted chain complex and based Reidemeister torsion for link exteriors

[F1]

The finite procedure checks its declared input conditions, evaluates $D$ before the acyclicity branch, returns zero for a nonacyclic complex, and otherwise returns $N / D$ . Compute the twisted Burau determinant for a colored braid closure

[F2]

For the certified closure and axis exteriors with $D \neq 0$ , the braid-axis torsion product identifies the exterior torsion with $N / D$ on the acyclic branch and proves $N = 0$ on the nonacyclic branch. The braid-axis torsion product gives the twisted Burau determinant

[F3]

The twisted torsion convention assigns zero to a nonacyclic fraction-field complex and takes acyclic torsion modulo the determinant and deck-unit subgroup. Twisted chain complex and based Reidemeister torsion for link exteriors

[F4]

The colored-braid algorithm forms the full positive-meridian abelianization before applying the declared color map. Compute the twisted Burau determinant for a colored braid closure

Proof

1.1A1givenF1F4

By [A1] and the given input hypotheses, all finite spine, gluing, color and representation checks in [F1] pass. Since $D \neq 0$ , the designated deleted column is admissible and the procedure does not return not-applicable. The coefficient system is the full positive-meridian system followed by the declared color map, as required by [F4], and the axis meridian acts trivially by the extension hypothesis.

2.1F1F2F3step 1.1cases∎

If the fraction-field complex is acyclic, [F1] returns $N / D$ and [F2] identifies this value with the twisted torsion class. If the complex is nonacyclic, [F1] returns $0$ , [F2] gives $N = 0$ , and [F3] assigns torsion $0$ . Thus in either branch the procedure returns the twisted Reidemeister torsion of the closure exterior, with precisely the stated spine and extension hypotheses.

Depends on

Used by

Trefoil torsion from its torus spine and colored Burau block Example

Dependency tree · two levels

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Sources

Anthony Conway, Burau Maps and Twisted Alexander Polynomials, Theorem 3.15 (standard reference, not scraped)
Paul Kirk and Charles Livingston, Twisted Alexander Invariants, Reidemeister Torsion, and Casson–Gordon Invariants, §§2–4 (standard reference, not scraped)
Andrew Ranicki, Notes on Reidemeister Torsion, Definitions 4 and 6 and Theorem 15 (standard reference, not scraped)