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Seifert block presentation of cyclic branched-cover homology

Statement

Assume the Axiom of Choice. Let $K$ be a smooth oriented knot, let $F$ be a connected compact oriented Seifert surface of genus $g$ , and choose any ordered integral basis $a_{1}, \dots, a_{2 g}$ of $H_{1} (F; Z)$ with positive-push Seifert matrix $V_{i j} = lk (a_{i}^{+}, a_{j})$ . For an integer $q \geq 2$ , let $M_{q} (V)$ be the symmetric block matrix of size $2 g (q - 1)$ whose $2 g \times 2 g$ block in position $(k, ℓ)$ , for $1 \leq k, ℓ \leq q - 1$ , is $V + V^{T}$ if $k = ℓ$ , $- V^{T}$ if $ℓ = k + 1$ , $- V$ if $k = ℓ + 1$ , and $0$ if $∣ k - ℓ ∣ \geq 2$ . Then

H_{1} (Σ_{q} (K); Z) ≅ coker (M_{q} (V)) .

For $q = 2$ this presentation matrix is $V + V^{T}$ ; when $g = 0$ the empty matrix has trivial cokernel.

Statement references

Connected compact oriented spanning surfaces and their genus

The positive-push Seifert matrix convention

Bilinearity and basis-change behavior of the Seifert pairing

The framed-link matrix of the branched 4-manifold

The exterior cover and filling that define $Σ_{q} (K)$

Poincaré–Lefschetz duality for the handlebody pair

Facts & Assumptions

Given: Full AC, the oriented knot $K$ , a connected compact oriented Seifert surface $F$ , an ordered integral basis of $H_{1} (F; Z)$ , its positive-push matrix $V$ , and $q \geq 2$ .

[A1]

Full AC is the choice-function axiom used by the cyclic-cover and duality suppliers. The Axiom of Choice

[F1]

The cover $Σ_{q} (K)$ is the boundary of the $q$ -fold branched cover of $B^{4}$ along a push-in of $F$ . Cyclic branched cover of a knot, Surgery presentation from a Seifert surface

[F2]

For the disk-band basis, that 4-manifold has one $0$ -handle and $2 g (q - 1)$ $2$ -handles, with intersection matrix $M_{q} (V)$ . Surgery presentation from a Seifert surface

[F3]

Poincaré–Lefschetz duality identifies $H^{2} (W; Z)$ with $H_{2} (W, \partial W; Z)$ by capping with the relative fundamental class. Poincaré–Lefschetz duality

[F4]

The homology long exact sequence of the pair $(W, \partial W)$ is exact. Long exact sequence of a pair

[F5]

The Seifert pairing is bilinear on homology classes; changing an integral basis by $P \in {GL}_{2 g} (Z)$ changes its matrix to $P^{T} V P$ . Seifert pairing and matrix of an oriented knot, Well-definedness of the Seifert pairing

Proof

Given: The data in the Statement and Facts & Assumptions. First use the disk-band basis from the surgery construction, and write $Y = \partial W = Σ_{q} (K)$ and $m = 2 g (q - 1)$ .

1.1A1F2F3givenalgebra

The handle decomposition in [F2] has cellular chain groups $C_{2} (W; Z) = Z^{m}$ , $C_{1} (W; Z) = 0$ , and $C_{0} (W; Z) = Z$ . Hence $H_{2} (W; Z) = Z^{m}$ , $H_{1} (W; Z) = 0$ , and $H^{2} (W; Z) = Z^{m}$ . By [F3], $H_{2} (W, Y; Z) = Z^{m}$ as well.

2.1F2F3step 1.1givenconstruct

In the handle bases, the map $j : H_{2} (W; Z) \to H_{2} (W, Y; Z)$ sends each absolute handle class to its intersection coordinates with the relative dual handle classes. A diagonal coordinate is the handle framing, and an off-diagonal coordinate is the linking of the corresponding attaching circles; by [F2] this matrix is $M_{q} (V)$ .

3.1F1F2F3F4step 1.1step 2.1algebra

The pair sequence [F4] contains $H_{2} (W; Z) \overset{j}{\to} H_{2} (W, Y; Z) ⟶ H_{1} (Y; Z) ⟶ H_{1} (W; Z) = 0$ . By [F1], $Y = \partial W = Σ_{q} (K)$ . Exactness and step 2.1 therefore give $H_{1} (Σ_{q} (K); Z) ≅ coker (M_{q} (V))$ in the disk-band basis.

4.1F5step 3.1algebra∎

For any other ordered integral basis, [F5] gives $V^{'} = P^{T} V P$ with $P$ unimodular. Let $P_{q} = diag (P, \dots, P)$ with $q - 1$ diagonal blocks. Blockwise substitution in the definition shows $M_{q} (V^{'}) = P_{q}^{T} M_{q} (V) P_{q}$ . Left and right multiplication by unimodular integer matrices preserve the isomorphism class of the cokernel, so the presentation holds in the stated basis. For $q = 2$ the block matrix has the single block $V + V^{T}$ , and for $g = 0$ it is empty.

Depends on

Used by

Homology of the double branched cover from a Seifert matrix Lemma

Dependency tree · two levels

59 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)
Jozef H. Przytycki and Akira Yasuhara, Linking Numbers in Rational Homology 3-Spheres, Cyclic Branched Covers and Infinite Cyclic Covers (standard reference, not scraped)