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The Khovanov to branched-cover Heegaard Floer spectral sequence

Statement

Assume AC and ACω. For every nonempty oriented link $L \subset S^{3}$ , there is a spectral sequence over $F_{2}$ whose $E_{2}$ page is the reduced Khovanov cohomology ${\tilde{K h}}^{*, *} (\overline{L}; F_{2})$ of the mirror link and whose abutment is $\hat{H F} (Σ (L); F_{2})$ with its image filtration. The page identification is an isomorphism of bigraded vector spaces; this statement does not claim a diagram-independent filtered quasi-isomorphism type for the chain-level spectral sequence.

Statement references

Fixed-diagram spectral sequence

Reduced Khovanov link invariance

Branched-cover invariance

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, Theorem 1.1, Corollary 1.2, and §6

Lipshitz, Heegaard Floer Homologies, §5.3

Facts & Assumptions

Given: AC, ACω, and a nonempty oriented link $L$ .

[F0]

Every nonempty oriented link in $S^{3}$ has a finite regular diagram: move it off $\infty$ using the link definition's ambient-isotopy convention, then choose a regular projection direction. Every component has a point away from the finite crossing set for the basepoint (Oriented links in the three-sphere and ambient isotopy, Existence of regular projections).

[F1]

The branched cover is well-defined up to oriented diffeomorphism (Existence and isotopy invariance of the double branched cover of a link).

[F2]

For a fixed diagram, the surgery construction has $E_{2}$ equal to based reduced Khovanov cohomology and converges to the branched-cover Floer group (The Ozsváth–Szabó spectral sequence for a fixed link diagram).

[F3]

Over ACω, the reduced Khovanov group is independent of the chosen diagram and marked component up to bigraded isomorphism (Reduced Khovanov homology over F2 is an oriented link invariant).

[A1]

AC is used by the link-complement and covering-space construction; ACω is separately used by the Reidemeister-move and generic Floer analytic branches (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F0F1F2A1givenchoose

By [F0], choose an ambient-isotopic representative $L^{'}$ with a finite regular diagram $D$ and a point on its mirror away from crossings. Apply [F2] to $D$ ; this gives a spectral sequence with $E_{2}$ equal to the based reduced Khovanov group of the mirror diagram and abutment $\hat{H F} (Σ (L^{'}); F_{2})$ . By [F1], $\hat{H F} (Σ (L^{'}); F_{2}) ≅ \hat{H F} (Σ (L); F_{2})$ .

2.1F1F2F3step 1.1

Apply [F3] to identify that page, up to bigraded isomorphism, with $\tilde{K h} (\overline{L}; F_{2})$ independently of the selected diagram and component. The convergence in [F2] and target invariance in [F1] give the stated invariant endpoints, with no diagram-independent filtered chain claim.

3.1F2F3step 1.1step 2.1∎

The two deductions above are exactly the stated endpoint construction relative to [F2] and [F3]. They preserve the fixed-diagram scope of the filtered construction and use only the invariant isomorphism classes of its endpoints. Thus the formal assembly is complete relative to these two exact supplier interfaces.

External-input boundary

The fixed-diagram supplier uses the explicitly labelled external Ozsváth–Szabó surgery and unlink-map inputs, with exact locators and reading limits recorded in those suppliers and the run source receipt. The reduced link-invariance supplier supplies the independent based-link passage. This proof assembles those results; it does not reconstruct their imported parts or claim a diagram-independent filtered chain type. AC for the branched cover and ACω for the analytic/Reidemeister branches remain the separate assumptions of the Statement.

Depends on

Used by

The trefoil Khovanov-to-Floer rank calculation Example

Dependency tree · two levels

39 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

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, Theorem 1.1, Corollary 1.2, and §6 (standard reference, not scraped)
Lipshitz, Heegaard Floer Homologies, §5.3 (standard reference, not scraped)