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Reduced Khovanov cohomology feeds the singular instanton spectral sequence

Statement

Assume full AC. For every smooth knot $K \subset S^{3}$ , there is a finite spectral sequence of abelian groups whose $E_{2}$ page, after forgetting the Khovanov bigrading, is $\tilde{K h} (\overline{K}; Z)$ and which converges to $I^{♮} (K; Z)$ . The intermediate pages are not asserted to be knot invariants.

Statement references

Fixed mirror involution

Integral reduced cohomology convention

Based reduced invariance

Diagramwise resolution filtration

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), Proposition 1.2 and Theorem 8.12

Kronheimer–Mrowka, Knots, three-manifolds and instantons (2018), Theorem 5.3

Yang, An Overview of Floer Homologies (2025), §5.7

Facts & Assumptions

Given: Full AC, a smooth knot $K \subset S^{3}$ , and any finite based regular diagram $D$ of $K$ .

[F1]

The fixed reflection defining the mirror satisfies $\overline{\overline{K}} = K$ (Mirror of a classical knot, Definition).

[F2]

Under AC $_{ω}$ , the reduced integral groups of based diagrams of the same oriented knot are isomorphic, and transport of the basepoint along the knot preserves the groups (Reduced Khovanov cohomology is invariant under based moves and basepoint transport, Statement).

[F3]

For the chosen diagram, the filtered instanton complex has $E_{2} = \tilde{K h} (\overline{D}; Z)$ and abuts to $I^{♮} (K; Z)$ (Filtered instanton complex from the cube of knot resolutions, Statement). Its actual proof supplies the integral multi-saddle counts, exterior-line signs, filtered cone comparisons and planar unlink operations.

Proof

1.1F3given

Apply [F3] to $D$ . This gives a finite spectral sequence from the reduced Khovanov cohomology of the reflected based diagram $\overline{D}$ to the singular instanton group of $K$ , with no assertion about invariance of the intervening pages.

2.1F1F2step 1.1algebra∎

The fixed reflection sends $D$ to a diagram of $\overline{K}$ and carries its marked point with it. By [F2], the reduced groups do not depend on the chosen based diagram of $\overline{K}$ . Thus the $E_{2}$ page from step 1.1 is the stated knot invariant $\tilde{K h} (\overline{K}; Z)$ , with its bigrading forgotten.

Depends on

Used by

Reduced Khovanov cohomology detects the unknot Theorem

Dependency tree · two levels

26 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

P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, Proposition 1.2 and Theorem 8.12 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Knots, three-manifolds and instantons, Theorem 5.3 (standard reference, not scraped)
Hongjian Yang, An Overview of Floer Homologies, §5.7 (standard reference, not scraped)