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The Seifert determinant is the knot Alexander polynomial

Statement

Assume the full Axiom of Choice. Let $K \subset S^{3}$ be a smooth oriented knot, and let $F \subset S^{3}$ be a connected compact smooth oriented embedded surface whose oriented boundary is $K$ , with $χ (F) = 1 - 2 g$ . Choose an ordered integral basis $a_{1}, \dots, a_{2 g}$ of $H_{1} (F; Z)$ and use the positive normal convention $V_{i j} = lk (a_{i}^{+}, a_{j})$ . Let $A_{K}$ be the first homology of the infinite cyclic cover of $S^{3} ∖ K$ classified by total linking, with $t$ acting by the positive-meridian deck transformation. Let $Δ_{K} (t)$ be the generator of the zeroth elementary ideal $E_{0} (A_{K})$ in $Λ = Z [t^{\pm 1}]$ , and write $p ≐ q$ when $p$ and $q$ differ by a unit $\pm t^{k}$ of $Λ$ . Then for every such $F$ and basis,

$\det (V - t V^{T}) ≐ Δ_{K} (t) .$

The determinant is $1$ for the empty $0 \times 0$ matrix when $g = 0$ . This formula concerns the polynomial $Δ_{K}$ ; the existing knot-normalized fraction $D_{K} = Δ_{K} / (1 - t)$ is a separate invariant.

Statement references

Connected compact oriented spanning surfaces and their genus

The positive-push Seifert matrix convention

The infinite cyclic cover and positive deck action

The gcd-E0 polynomial and the separate knot-normalized fraction

The square presentation's E0 generator and evaluation at one

Facts & Assumptions

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

[A1]

Full AC is the choice-function axiom; it is the stated hypothesis of the Alexander-module and Seifert-matrix presentation suppliers. (The Axiom of Choice)

[F7]

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

[F1]

The surface has the specified oriented boundary, and its positive-push matrix is $V_{i j} = lk (a_{i}^{+}, a_{j})$ . (Oriented Seifert surfaces in the three-sphere, Seifert pairing and matrix of an oriented knot)

[F2]

With the same positive normal and positive-meridian deck convention, $A_{K} ≅ coker (V - t V^{T})$ over $Λ$ . (A Seifert matrix presents the one-variable Alexander module)

[F3]

For every such matrix, $E_{0} (A_{K}) = (\det (V - t V^{T}))$ and $\det (V - V^{T}) = \pm 1$ . (The square Seifert presentation generates the knot Alexander polynomial)

[F4]

$Δ_{K}$ is a generator of $E_{0} (A_{K})$ ; $Λ = Z [t^{\pm 1}]$ is an integral domain whose units are exactly $\pm t^{k}$ . (The one-variable Alexander module of an oriented link, The Alexander polynomial from the zeroth elementary ideal, The Laurent polynomial ring as the principal localisation of Z[t] at t, Units, powers and the domain property of the Laurent polynomial ring)

[F5]

Under an orientation-preserving ambient isotopy of knot-surface pairs, the transported Seifert matrix is unchanged and any other target basis changes it by $V^{'} = P^{T} V P$ with $P \in {G L}_{2 g} (Z)$ . (Seifert matrices of ambient-isotopic pairs are S-equivalent)

[F6]

The Alexander modules of orientation-preserving ambient-isotopic links are isomorphic, so their Alexander polynomials agree up to a unit. (The Alexander polynomial is an oriented link invariant)

Proof

1.1A1F7F2F3F4givenalgebra

Fix the arbitrary knot, surface, and basis in the Statement, put $n = 2 g$ , and set $M = V - t V^{T}$ . By [F2] the cokernel of $M$ is $A_{K}$ , and [F3] gives $E_{0} (A_{K}) = (\det M)$ . By [F4], $Δ_{K}$ also generates this ideal. The local lemma gives $(\det M) (1) = \pm 1$ , so $\det M \neq 0$ . If $Δ_{K} = a \det M$ and $\det M = b Δ_{K}$ for $a, b \in Λ$ , then $\det M = b a \det M$ ; since $Λ$ is a domain and $\det M \neq 0$ , $b a = 1$ . Thus $a$ is a unit, and [F4] identifies it as $\pm t^{k}$ . Hence $\det M ≐ Δ_{K}$ .

2.1F1F5F6step 1.1givenalgebra∎

If the basis is changed by $P \in {G L}_{n} (Z)$ , then $M^{'} = P^{T} M P$ and $\det M^{'} = (\det P)^{2} \det M = \det M$ . Under an orientation-preserving ambient isotopy, [F5] gives the same transported-matrix relation and [F6] preserves $Δ_{K}$ up to a unit. Since the original $K, F$ , and basis were arbitrary, the formula holds for every surface and basis in the Statement and is compatible with the oriented knot invariant.

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
The square knot is slice despite its nonunit Alexander polynomial Counterexample
The trefoil fails both algebraic sliceness tests Counterexample
S-equivalence and Conway normalization Definition
The square-knot Seifert form is metabolic Example
Trefoil and figure-eight Alexander polynomials from Seifert matrices Example
Unknot, trefoil, and positive untwisted Whitehead double cases Example
An Alexander-one Seifert form is S-equivalent to the empty form Lemma
Root-product test for cyclic branched-cover homology Lemma
The Conway knot has normalized Alexander polynomial one Lemma
The Seifert determinant detects Arf parity near minus one Lemma
The singular parameters of the Levine–Tristram form Lemma
A metabolizer forces Levine–Tristram signatures to vanish Theorem
Infinitely many SL2(C)-abelian surgeries characterize torus knots Theorem
Knot determinant is the order of double-cover homology Theorem
Locally flat sliceness implies Fox–Milnor factorization Theorem
Seifert S-equivalence and the Conway–Arf bridge Theorem
The enhanced A-polynomial Newton polygon detects torus knots Theorem
Torus-knot Alexander polynomial and genus formulas Theorem

Dependency tree · two levels

73 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)