TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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 Alexander polynomial is an oriented link invariant

Statement

Assume the Axiom of Choice. Let $L$ and $L^{'}$ be nonempty oriented links in $S^{3}$ such that some ambient isotopy of $S^{3}$ carries $L$ onto $L^{'}$ and preserves the orientations of the components (Oriented links in the three-sphere and ambient isotopy). Then the Alexander modules $A_{L}$ and $A_{L^{'}}$ of The one-variable Alexander module of an oriented link are isomorphic as $Λ$ -modules over $Λ = Z [t^{\pm 1}]$ , their zeroth elementary ideals agree, $E_{0} (A_{L}) = E_{0} (A_{L^{'}})$ , and consequently the one-variable Alexander polynomials of The Alexander polynomial from the zeroth elementary ideal satisfy $Δ_{L^{'}} (t) = \pm t^{k} Δ_{L} (t) for some k \in Z;$ so the oriented link type of $L$ determines $Δ_{L}$ up to multiplication by a unit $\pm t^{k}$ of $Λ$ (and, for knots, the Alexander invariant $D_{L} = Δ_{L} / (1 - t)$ up to the same unit).

Facts & Assumptions

Given: AC and oriented links $L$ and $L^{'}$ in $S^{3}$ with an ambient isotopy carrying $L$ onto $L^{'}$ and preserving orientations.

[A1]

The Axiom of Choice, used only through the definition of the Alexander module, which uses Alexander duality to build the total linking homomorphism (The Axiom of Choice, The one-variable Alexander module of an oriented link).

[F1]

The given ambient isotopy $H : S^{3} \times I \to S^{3}$ has $H_{0} = i d$ and $H_{1} (L) = L^{'}$ . Its time-one map $h : = H_{1}$ is an orientation-preserving diffeomorphism of $S^{3}$ carrying $L$ onto $L^{'}$ with component orientations preserved (Oriented links in the three-sphere and ambient isotopy).

[F2]

For an oriented link $M$ the total linking homomorphism is the composite $H_{1} (X_{M}; Z) \tilde{\to} {\tilde{H}}^{1} (M; Z) ≅ ⨁_{i} H^{1} (K_{i}; Z) ≅ Z^{# π_{0} (M)} \overset{\sum}{\to} Z$ induced by Alexander duality and the orientation-induced identifications; the cover $\tilde{X_{M}}$ is the connected infinite cyclic cover classified by $K = \ker (π_{1} (X_{M}) \to H_{1} (X_{M}) \overset{φ}{\to} Z)$ , it is regular with deck group $Z$ , and $A_{M} = H_{1} (\tilde{X_{M}}; Z)$ is a $Λ$ -module with $t$ acting by the positive deck transformation (The one-variable Alexander module of an oriented link).

[F3]

Elementary ideals of a finitely presented module depend only on the isomorphism class of the module, not on its presentation (Elementary ideals are independent of the presentation, The one-variable Alexander module of an oriented link).

[F4]

$Λ = Z [t^{\pm 1}]$ is a unique factorisation domain whose units are exactly $\pm t^{k}$ , $k \in Z$ (The Laurent polynomial ring is Noetherian and a unique factorisation domain, Units, powers and the domain property of the Laurent polynomial ring); the polynomial $Δ_{M}$ is a gcd of $E_{0} (A_{M})$ and is well defined up to multiplication by such a unit (The Alexander polynomial from the zeroth elementary ideal).

[F5]

Cap naturality holds on chains and therefore on compact-support classes (Cap naturality and projection formula). Pair connectors, restriction and excision commute with a homeomorphism by their chain/cochain definitions. Thus the Alexander-duality construction in its supplier’s Proof 4.1–6.1 commutes with orientation-preserving ambient homeomorphisms: the fundamental class is preserved and the cap-natural diagram, followed by the natural pair connector, gives the duality-natural diagram.

[F6]

A based map lifts when it carries the source covering subgroup into the target subgroup (Lifting criterion for maps from path-connected locally path-connected spaces); two connected-domain lifts agreeing at a point coincide (Two lifts from a connected space that agree at one point agree everywhere).

Proof

1.1F1given

The ambient homeomorphism. By [F1] the ambient isotopy carrying $L$ onto $L^{'}$ restricts to a homeomorphism $h : S^{3} \to S^{3}$ with $h (L) = L^{'}$ ; restricting $h$ to the complements gives a homeomorphism $X_{L} \to X_{L^{'}}$ of the link complements, and it is orientation preserving because it is the time-one map of an isotopy of $S^{3}$ .

2.1F2F5F6step 1.1

Transport of the linking homomorphisms. The homeomorphism carries each component $K_{i}$ of $L$ onto the corresponding component $K_{i}^{'}$ of $L^{'}$ and preserves the orientations, hence it carries the orientation-induced generator of $H^{1} (K_{i}; Z)$ to the corresponding generator of $H^{1} (K_{i}^{'}; Z)$ ; by the Alexander-duality description [F2] and the naturality justified in [F5], the induced isomorphism $h_{*} : H_{1} (X_{L}; Z) \to H_{1} (X_{L^{'}}; Z)$ satisfies $φ_{L^{'}} \circ h_{*} = φ_{L}$ . Consequently $h_{*} (\ker φ_{L}) = \ker φ_{L^{'}}$ , and the $Z$ -cover classified by $\ker φ_{L^{'}}$ pulls back along $h$ to a cover of $X_{L}$ isomorphic to $\tilde{X_{L}}$ ; by [F6] choose a normalized lift of $h$ and the corresponding lift of $h^{- 1}$ . Their composites are normalized lifts of the identity and therefore are the identity. For a loop with total linking number $1$ , lifting the loop and its $h$ -image shows that the lift sends the level-one fibre point to the level-one point. Hence its two composites with the positive deck generators agree there, and [F6] makes them equal everywhere. The lift is therefore a deck-equivariant homeomorphism.

3.1A1F2F3step 2.1

Module isomorphism and elementary ideals. The deck-equivariant lift of $h$ induces a $Λ$ -module isomorphism $A_{L} = H_{1} (\tilde{X_{L}}; Z) ≅ H_{1} (\tilde{X_{L^{'}}}; Z) = A_{L^{'}}$ , since it intertwines the deck actions and the identification $t \leftrightarrow$ positive deck transformation [F2]. By [F3] elementary ideals are invariants of the isomorphism class, so $E_{0} (A_{L}) = E_{0} (A_{L^{'}})$ as ideals of $Λ$ .

4.1F4step 3.1algebra∎

The polynomial. Both $Δ_{L}$ and $Δ_{L^{'}}$ are gcds of the same ideal $E_{0} (A_{L}) = E_{0} (A_{L^{'}})$ by [F4], and in a unique factorisation domain two gcds of the same set of elements differ by a unit; since the units of $Λ$ are exactly $\pm t^{k}$ by [F4], this gives $Δ_{L^{'}} = \pm t^{k} Δ_{L}$ . The same computation applies to the knot normalisation $D_{M} = Δ_{M} / (1 - t)$ , whose unit ambiguity is that of $Δ_{M}$ .

Remarks

The theorem is the reason the Burau determinant of The Burau determinant recovers the Alexander polynomial of a closed braid computes an oriented link invariant from any braid representative: the right-hand side is computed for one representative and the left-hand side is the invariant supplied here.
The Axiom of Choice enters only through the Alexander module; no additional choice is made in the proof. The ambient isotopy is already given, so no isotopy-extension theorem is needed.

Depends on

Used by

The Burau determinant recovers the Alexander polynomial of a closed braid Proposition

Dependency tree · two levels

63 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

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.2 (printed pp. 46-47) and section 4.4 (printed p. 52): invariance of the Alexander polynomial under ambient isotopy (standard reference, not scraped)
R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and Co. (1963), chapters VIII-IX (the Alexander module and its invariance) (standard reference, not scraped)