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Homology of the double branched cover from a Seifert matrix

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 let $V$ be its positive-push Seifert matrix in any ordered integral basis of $H_{1} (F; Z)$ . Then

H_{1} (Σ_{2} (K); Z) ≅ coker (V + V^{T}) .

For $g = 0$ , the matrix is empty and the cokernel is trivial.

Statement references

The definition of the double branched cover $Σ_{2} (K)$

The connected compact oriented spanning surface and genus convention

The positive-push Seifert matrix convention

The cyclic-cover homology presentation by $M_{q} (V)$

Rolfsen, Knots and Links, Chapter 8.D

Facts & Assumptions

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

[A1]

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

[F1]

The double branched cover is $Σ_{2} (K)$ from the connected meridian-mod- $2$ exterior cover with its lifted meridian filled. Cyclic branched cover of a knot

[F2]

For every $q \geq 2$ , $H_{1} (Σ_{q} (K); Z) ≅ coker (M_{q} (V))$ , with diagonal blocks $V + V^{T}$ and adjacent blocks $- V$ and $- V^{T}$ . Seifert block presentation of cyclic branched-cover homology

Proof

1.1givenF2algebra

The block indices in $M_{q} (V)$ run from $1$ to $q - 1$ . At $q = 2$ there is only one index, so there are no adjacent blocks and $M_{2} (V)$ consists of the single diagonal block $V + V^{T}$ .

2.1A1F1F2step 1.1given∎

Substitute $q = 2$ in [F2]. By [F1], the resulting manifold is the double branched cover $Σ_{2} (K)$ ; hence its first homology is the cokernel of $V + V^{T}$ . If $g = 0$ , the block has size zero and the cokernel is the zero group by the empty-matrix convention. The required AC hypothesis is inherited through [F1].

Depends on

Used by

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Sources

Dale Rolfsen, Knots and Links (standard reference, not scraped)
Charles Livingston, Seifert Forms and Concordance (standard reference, not scraped)