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Surgery presentation from a Seifert surface

Statement

Assume the Axiom of Choice. Let $K$ bound a connected compact oriented Seifert surface $F$ of genus $g$ . Choose a disk-band decomposition of $F$ with one disk and $2 g$ bands, and use its band-core basis $a_{1}, \dots, a_{2 g}$ to form the positive-push Seifert matrix $V_{i j} = lk (a_{i}^{+}, a_{j})$ . For $q \geq 2$ , define the symmetric block matrix $M_{q} (V)$ of size $2 g (q - 1)$ by specifying its $2 g \times 2 g$ blocks, indexed by $1 \leq k, ℓ \leq q - 1$ : $(M_{q} (V))_{k ℓ} = {\begin{cases} V + V^{T}, & k = ℓ, \\ - V^{T}, & ℓ = k + 1, \\ - V, & k = ℓ + 1, \\ 0, & ∣ k - ℓ ∣ \geq 2. \end{cases}$ Push the interior of $F$ into $B^{4}$ while keeping $K$ on $S^{3} = \partial B^{4}$ , and let $W_{q} (F)$ be the actual $q$ -fold cyclic branched cover along that pushed-in surface. Use the following disk-band handle model of this cover. Cut $B^{4}$ along the three-dimensional track of the push-in isotopy; the cut-open manifold is again a $4$ -ball and its boundary contains the thickened surface $\overset{ˉ}{F} = (F \times [- 1, 1]) / \sim$ , where $(x, t) \sim (x, t^{'})$ for $x \in \partial F$ . Write ${\overset{ˉ}{F}}^{\pm} = {(x, t) \in \overset{ˉ}{F} : \pm t \geq 0}$ . Take $q$ copies of this cut-open ball and, for $i = 1, \dots, q - 1$ , glue ${\overset{ˉ}{F}}_{i}^{+}$ to ${\overset{ˉ}{F}}_{i + 1}^{-}$ by $(x, t) \mapsto (x, - t)$ . The Akbulut–Kirby construction identifies this handle model with $W_{q} (F)$ ; the last-to-first gluing is unnecessary for the underlying manifold in this model. This is a statement about the isotopy-track handle model, not an assertion that an open chain of normal-disk sectors is already the cyclic covering map. The resulting handle presentation has one $0$ -handle and $2 g (q - 1)$ $2$ -handles, and its framed attaching link has linking matrix $M_{q} (V)$ . The boundary branched cover, with the boundary marking induced by the covering map of $W_{q} (F)$ , is $Σ_{q} (K)$ .

Statement references

Connected compact oriented spanning surfaces and the genus convention

The positive-push Seifert matrix convention

The disk-band handle form for a compact surface with one boundary component

The exterior cover and meridional filling defining $Σ_{q} (K)$

Akbulut–Kirby, “Branched Covers of Surfaces in 4-Manifolds,” §2

Przytycki–Yasuhara, §3

Facts & Assumptions

Given: Full AC, the oriented knot $K$ , a connected compact oriented Seifert surface $F$ , its disk-band decomposition, the ordered band-core basis, and $q \geq 2$ .

[A1]

Full AC implies the countable-choice hypothesis of the finite surface normal-form supplier. The Axiom of Choice, AC implies DC implies countable choice

[F1]

A Seifert surface is connected, compact, smooth, oriented, and embedded with oriented boundary $K$ ; its genus is the integer $g$ with $χ (F) = 1 - 2 g$ . Oriented Seifert surfaces in the three-sphere

[F2]

Under countable choice, a connected compact surface with one boundary component has a disk-band handle form with $2 g$ bands and a standard integral homology basis given by the band cores. Finite surface normal forms, Jordan disks, and torsion control

[F3]

The positive-push Seifert matrix has entries $V_{i j} = lk (a_{i}^{+}, a_{j})$ , with the positive normal determined by the surface and ambient orientations. Seifert pairing and matrix of an oriented knot

[F4]

The $q$ -fold branched cover of $S^{3}$ is formed from the meridian-mod- $q$ cover of the compact exterior by filling the lift of $μ^{q}$ . Cyclic branched cover of a knot

[F5]

External input (Akbulut–Kirby disk-band construction). For a connected oriented Seifert surface in $S^{3}$ , pushed into $B^{4}$ , the $q$ -fold cyclic branched cover is represented by cutting along the push-in isotopy track and gluing $q$ cut-open balls along the thickened half-surfaces as specified above, using only the $q - 1$ adjacent-copy gluings. The terminal gluing is unnecessary for the underlying manifold. If the surface has one disk and $b$ bands, the disk gluings give one $0$ -handle; each band gives one $2$ -handle at each of the $q - 1$ layers, with attaching circle obtained from the two corresponding face-core arcs and with the carried band framing. The actual branched covering map restricts on the boundary to the cyclic cover branched along the boundary link. This is the explicitly imported construction of Akbulut–Kirby, §2, printed pp.113–114 and the boundary-restriction Remark p.116; the paper's smooth oriented conventions are on p.111. It is not claimed as a locally reconstructed handle-identification proof. Akbulut–Kirby, Branched Covers of Surfaces in 4-Manifolds, Math. Ann.252 (1980),111–131.

Proof

Given: Full AC, the oriented knot $K$ , the connected compact oriented Seifert surface $F$ , its disk-band decomposition and band-core basis, the positive-push matrix $V$ , and $q \geq 2$ .

1.1A1F1F2F5givenconstruct

Use the isotopy-track handle model supplied by [F5], not a cut of the codimension-two branch surface itself. By [F1] and [F2], the connected surface has one disk and $2 g$ bands. In the model, the disk-face gluings combine the $q$ cut-open $4$ -balls into a $4$ -ball with one $0$ -handle. The remaining face gluings give, for each band and each adjacent-copy layer $k = 1, \dots, q - 1$ , a $2$ -handle whose attaching circle joins that band's two face-core arcs through the common disk region, with the carried band framing. Hence there are $2 g (q - 1)$ $2$ -handles and no other handles. The identification of this model with the actual cyclic branched cover, including the legitimacy of omitting the terminal gluing, is the external input [F5]; the local linking-entry computation below uses its attaching-circle description.

2.1F3step 1.1givenconstructalgebra

For band $a_{i}$ and seam $k$ , the attaching circle $C_{i, k}$ is obtained by joining its oriented core arc in the positive face of copy $k$ to the core arc in the negative face of copy $k + 1$ , with endpoints joined through the common disk $0$ -handle. In the collar of $F$ , closing either face arc through that disk gives the corresponding positive or negative push-off of the closed core cycle $a_{i}$ . Simultaneously moving $a_{i}^{-}$ and $a_{j}$ through the product collar to $a_{i}$ and $a_{j}^{+}$ preserves linking, so symmetry of linking and [F3] give $lk (a_{i}^{-}, a_{j}) = lk (a_{j}^{+}, a_{i}) = V_{j i}$ . For two handles at the same seam, the positive-face and negative-face contributions are therefore $V_{i j}$ and $V_{j i}$ , giving $V_{i j} + V_{j i}$ ; for the self-entry this is framing $2 V_{i i}$ . For handles at consecutive seams, only their common sheet contributes, and the reflection reverses one normal direction, giving $- lk (a_{j}^{+}, a_{i}) = - V_{j i}$ in the upper block; symmetry gives $- V_{i j}$ in the lower block. If the seams are separated by at least one unused seam, the attaching circles lie in disjoint sheet neighborhoods, so their linking is zero. These cases exhaust all pairs and give exactly $M_{q} (V)$ .

2.2F4F5step 1.1given

Apply the branched covering map on the actual cover $W_{q} (F)$ identified with the handle model by [F5]. Its normal-disk map is $w \mapsto w^{q}$ ; this is not a claim about the unmatched terminal banks of the handle model. A positive meridian away from the branch locus moves cyclically through the $q$ sheets. Restriction to the boundary therefore gives the connected exterior cover classified by $π_{1} (E_{K}) \to Z / q$ , $μ \mapsto 1$ , and the lift of $μ^{q}$ bounds the normal filling disk. This is exactly the boundary restriction in [F5] and the exterior/filling definition [F4]. Thus the boundary with the marking induced by the actual branched covering map is $Σ_{q} (K)$ . No canonical covering map on the unclosed raw handle model is being asserted.

3.1step 2.1givenalgebra∎

For $q = 2$ there is one seam, so the block matrix has only its diagonal block $V + V^{T}$ , as stated.

Remarks

The boxes represent the $q$ copies of the cut $4$ -ball; each arrow represents the $2 g$ handles attached at one seam. The upper and lower matrix blocks record the opposite normal directions across that seam. The display sketches $q \geq 4$ ; when $q = 2$ there is one arrow. The proof computes the framed-link entries directly and does not rely on the schematic.

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Sources

Selman Akbulut and Robion Kirby, Branched Covers of Surfaces in 4-Manifolds (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)