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The Alexander polynomial from the zeroth elementary ideal

Definition

Assume the Axiom of Choice, so that the Alexander module $A_{L}$ of an oriented nonempty link $L$ is defined and finitely presented over $Λ = Z [t^{\pm 1}]$ (The one-variable Alexander module of an oriented link, The Alexander module of a link complement is finitely presented). Let $r$ be the number of components of $L$ . Define the one-variable Alexander polynomial $Δ_{L} (t) \in Λ$ to be a greatest common divisor of the zeroth elementary ideal $E_{0} (A_{L})$ (Elementary ideals of a finitely presented module), i.e. a generator of the smallest principal ideal of $Λ$ containing $E_{0} (A_{L})$ (The Laurent polynomial ring is Noetherian and a unique factorisation domain, Unique factorisation domain); it is well defined up to multiplication by a unit $\pm t^{k}$ of $Λ$ (Units, powers and the domain property of the Laurent polynomial ring). The Alexander invariant is the normalisation $D_{L} (t) : = {\begin{cases} Δ_{L} (t), & r > 1, \\ \frac{Δ_{L} (t)}{1 - t}, & r = 1, \end{cases}$ understood for a knot as an element of the fraction field of $Λ$ when $(1 - t) ∤ Δ_{L} (t)$ ; for the integer-valued link formulas of this page the distinction matters and is stated with each use.

Caveats. The ideal $E_{0} (A_{L})$ is a well-defined invariant of $L$ because $A_{L}$ is finitely presented and elementary ideals are independent of the presentation (Elementary ideals are independent of the presentation); a greatest common divisor in a unique factorisation domain is well defined up to units, and the units of $Λ$ are $\pm t^{k}$ . For every link the absolute one-variable Alexander module admits a square presentation, so $E_{0} (A_{L})$ is principal, generated by its determinant (the standard link fact recorded in [F3]). For a knot ( $r = 1$ ), every gcd representative satisfies $Δ_{L} (1) = \pm 1$ . These standard presentation and normalisation facts are recorded here; the gcd definition itself uses only finite presentability and unique factorisation.

Facts & Assumptions

Given: AC and an oriented link $L$ with $r$ components, its Alexander module $A_{L}$ over $Λ = Z [t^{\pm 1}]$ and its zeroth elementary ideal $E_{0} (A_{L})$ . No other choice principle is used.

[F1]

$A_{L}$ is a finitely generated $Λ$ -module, hence finitely presented, and $E_{0} (A_{L})$ is therefore defined and independent of the presentation (The Alexander module of a link complement is finitely presented, Elementary ideals of a finitely presented module, Elementary ideals are independent of the presentation).

[F2]

$Λ$ is a unique factorisation domain with units exactly $\pm t^{k}$ , $k \in Z$ ; in a unique factorisation domain a greatest common divisor of a nonempty set of elements exists and is well defined up to multiplication by a unit (The Laurent polynomial ring is Noetherian and a unique factorisation domain, Unique factorisation domain, Units, powers and the domain property of the Laurent polynomial ring).

[F3]

Literature input. For every oriented link, the absolute one-variable Alexander module has a square presentation (Burde–Zieschang, section 9.18, pp. 135–136); the connected-Seifert-surface presentation is $V^{T} - t V$ (Exercise 9.5, p. 140). Thus its zeroth elementary ideal is the principal determinant ideal, including the zero ideal when the determinant vanishes. For a knot with a Seifert matrix $V$ , the square matrix $V^{T} - t V$ presents the absolute Alexander module (Burde–Zieschang, Theorem 8.8, p. 110). Its determinant at $t = 1$ is $1$ in their canonical surface basis (Proposition 8.11, p. 112); hence $E_{0} (A_{L})$ is principal and every gcd representative has $Δ_{L} (1) = \pm 1$ . The $E_{0}$ indexing is essential because $A_{L}$ is absolute homology: for $A_{L} = Λ / (p)$ one has $E_{0} = (p)$ and $E_{1} = Λ$ . This is the standard normalisation of the Alexander polynomial (Burde–Zieschang, Theorem 8.8 and Proposition 8.11; locators in the references); it is recorded here and not used in the proofs of this page.

Proof

1.1F1F2

Well-definedness of $Δ_{L}$ . By [F1] the ideal $E_{0} (A_{L}) \subseteq Λ$ is an invariant of $L$ and $Λ$ is a UFD; write $E_{0} (A_{L}) = (g_{1}, \dots, g_{m})$ for a finite nonempty generating family (possible since $Λ$ is Noetherian; use the single generator $0$ for the zero ideal) and choose a greatest common divisor $g$ of $g_{1}, \dots, g_{m}$ , whose existence in a UFD is [F2]. Then $(g)$ is the smallest principal ideal containing $E_{0} (A_{L})$ : it contains every $g_{i}$ , hence $E_{0}$ , and any principal ideal $(h) \supseteq E_{0}$ contains all $g_{i}$ , so $h ∣ g$ by the defining property of the gcd, hence $(g) \subseteq (h)$ . Replacing $g$ by a unit multiple $\pm t^{k} g$ gives the same ideal, and by [F2] these are exactly the other choices. The normalisation $D_{L}$ is then defined by the displayed formula, with the fraction understood in the fraction field of the domain $Λ$ when $r = 1$ and $(1 - t) ∤ Δ_{L} (t)$ .

2.1F2F3step 1.1∎

The knot normalisation. For $r = 1$ the quoted standard fact [F3] identifies the normalisation of the Alexander polynomial used in the Burau comparison: $D_{L} = Δ_{L} / (1 - t)$ with $Δ_{L} (1) = \pm 1$ for every unit choice; for $r > 1$ no division is performed and $D_{L} = Δ_{L}$ . Both statements are part of the definition of the invariant used on this page.

Depends on

Used by

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.2 equation (15) and section 4.4 (the Alexander polynomial, its normalisation and the Burau determinant formula) (standard reference, not scraped)
Burde and Zieschang, Knots, second edition (2003), Theorem 8.8, Proposition 8.11, section 9.18 and Exercise 9.5, printed pp. 110, 112, 135-136 and 140 (absolute one-variable square presentations for knots and links; knot determinant at t=1) (standard reference, not scraped)
John Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (1968), 115-133 (the Alexander module, its elementary ideals and the Alexander polynomial) (standard reference, not scraped)