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Reduced Khovanov cohomology detects the unknot

Statement

Assume full AC. Let $K \subset S^{3}$ be a smooth knot. The total normalized integral reduced Khovanov cohomology is isomorphic to $Z$ concentrated in bidegree $(0, 0)$ if and only if $K$ is the unknot.

Statement references

Normalized integral reduced cohomology

Based reduced invariance

Reduced instanton spectral sequence

Sutured instanton rank detector

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), Theorem 1.1, Corollary 1.3, and Theorem 8.12

Kronheimer–Mrowka, Knots, three-manifolds and instantons (2018), Corollary 5.4

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

Facts & Assumptions

Given: Full AC, a smooth knot $K \subset S^{3}$ , and the normalized reduced integral Khovanov cohomology of $K$ .

[F1]

The fixed reflection satisfies $\overline{\overline{K}} = K$ and carries the standard unknot to itself (Mirror of a classical knot, Definition).

[F2]

For the crossingless based unknot diagram, the reduced quotient removes the based-circle $X$ label and places the surviving $1$ generator in bidegree $(0, 0)$ (Reduced Khovanov cohomology of a knot, Definition).

[A1]

For every knot $J$ , there is a finite spectral sequence from $\tilde{K h} (\overline{J}; Z)$ to $I^{♮} (J; Z)$ (Reduced Khovanov cohomology feeds the singular instanton spectral sequence, Statement). The integral resolution/cube construction is the retained original-facing completed supplier; it does not include the KHI rank implication.

[A2]

For every knot $J$ , $I^{♮} (J; Q) ≅ K H I (J; Q)$ and $\dim_{Q} K H I (J; Q) \geq 1$ , with equality exactly for the unknot (Sutured instanton knot homology detects the unknot, Statement). The supplier now records the needed ordinary surface, closure/excision and genus/nonvanishing results as explicitly published external inputs under the run-specific permission, retaining its local singular and unknot calculations. This does not pretend that the analytic complex or integral cube alone proves rank detection.

[A3]

The reduced groups are independent of the based diagram and basepoint under AC $_{ω}$ (Reduced Khovanov cohomology is invariant under based moves and basepoint transport, Statement). This is consumed at its declared mathematical interface; workflow readiness or historical owner hashes are not mathematical premises of this proof.

Proof

1.1A1F1A3givenconstruct

Suppose $\tilde{K h} (K; Z) ≅ Z$ as a total graded group. Set $J = \overline{K}$ . By [F1], $\overline{J} = K$ . The $E_{2}$ group in [A1] for $J$ is therefore $Z$ , so its rationalization has dimension one.

2.1A1step 1.1algebra

Put $S = Z ∖ {0}$ . Localization $S^{- 1} (-) = (-) \otimes_{Z} Q$ preserves exact sequences: if $f : A \to B$ is injective and $f (a) / s = 0$ , some $t \in S$ has $f (t a) = 0$ , hence $t a = 0$ and $a / s = 0$ ; if $g : B \to C$ follows $f$ and $g (b) / s = 0$ , choose $t \in S$ with $t b \in \ker g = im f$ , say $f (a) = t b$ , so $b / s = f (a) / (t s)$ ; if $g$ is surjective, each $c / s$ is the image of a lift divided by $s$ . Thus localization commutes with kernels, images, and homology.

3.1A1step 2.1algebra

The spectral sequence in [A1] is obtained by taking successive homology pages of a finite filtered complex. Exact localization therefore identifies its rationalized pages with the pages of the rationalized filtered complex, and identifies the localized abutment with $I^{♮} (\overline{K}; Q)$ . The $E_{2}$ page has rational dimension one by step 1.1.

4.1A1step 3.1algebra

On each finite-dimensional rational page, the next page is homology. For any differential $d$ on a finite-dimensional vector space $V$ , rank-nullity gives $\dim H (V, d) = \dim V - 2 \dim im d \leq \dim V$ . Iterating this inequality over the finite spectral sequence bounds the total dimension of $E_{\infty}$ by its one-dimensional $E_{2}$ page; since the finite abutment filtration has associated graded $E_{\infty}$ , its total dimension equals $\dim_{Q} I^{♮} (\overline{K}; Q) \leq 1$ .

5.1A2F1step 4.1algebra

By [A2], this target has dimension at least one, so it has dimension one. The rank-detection clause of [A2] implies that $\overline{K}$ is the unknot; applying the same reflection gives that $K$ is the unknot.

6.1F2givenalgebra∎

Conversely, for the crossingless one-circle unknot diagram the reduced complex has the single generator from [F2] and no differential. Its total cohomology is therefore $Z$ in bidegree $(0, 0)$ . This proves the converse and hence the stated equivalence.

Proof status

The rank argument is complete relative to the explicitly labelled supplier interfaces [A1]–[A3]. The accepted integral resolution construction supplies [A1]; its proof is reused without rereading or revalidating it. The rank detector [A2] now supplies its precise published ordinary-surface, closure/excision and genus/nonvanishing inputs under the expanded run-specific permission; this is a source-qualified use, not a claim of local reconstruction of those inputs. With [A1]–[A3], the finite localization/rank argument proves the unchanged Statement. It uses no distinguished instanton vector and gives no workflow-readiness conclusion.

Depends on

Used by

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Sources

P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, Theorem 1.1, Corollary 1.3, and Theorem 8.12 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Knots, three-manifolds and instantons, Corollary 5.4 (standard reference, not scraped)
Hongjian Yang, An Overview of Floer Homologies, §5.7 (standard reference, not scraped)