DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Odd Khovanov–Bar–Natan–LEO triple

Definition

Let $D$ be a finite regular oriented link diagram with at least one component, ordered crossings $c_{1} < \dots < c_{n}$ , and a marked point $p$ on the projected link away from crossings. Put $A = Z [X] / (X^{2})$ and $F_{2} = Z / 2 Z$ . At a resolution state $v \in {0, 1}^{n}$ , let $V_{v}$ be the free abelian group generated by the resolution circles, order its circle generators, and use the exterior algebra $Λ^{*} V_{v}$ as its state module. A wedge of $r$ distinct circle generators has circle quantum degree $k (v) - 2 r$ , where $k (v) = rank V_{v}$ ; its total normalized degrees are $i (v) = ∣ v ∣ - n_{-}$ and $q = ∣ v ∣ + n_{+} - 2 n_{-} + k (v) - 2 r$ . For a merge saddle joining circles $C_{1}, C_{2}$ , let $p : V_{v} \to V_{v + e}$ be the quotient sending both to the merged circle and fixing the other generators; the edge map is $Λ^{*} p$ . For a split saddle replacing $C$ by $C_{1}, C_{2}$ , let $j : V_{v} \to V_{v + e}$ send $C$ to $C_{1}$ and fix the other generators; the unsigned edge map is $z \mapsto (C_{1} - C_{2}) \land Λ^{*} j (z)$ . Choosing $C_{2}$ instead of $C_{1}$ gives the same map, while reversing the order of $C_{1}, C_{2}$ changes only its sign. Multiply each edge map by a sign $ϵ (e) \in {1, - 1}$ , incorporating any circle-generator reordering sign. The direct sum of these state modules with the resulting cube differential is the odd Khovanov cochain complex $C_{o} (D)$ . Write $Z_{p}$ for the circle containing the mark. Use the right action $z \cdot X = z \land Z_{p}$ , making $C_{o} (D)$ a complex of right $A$ -modules; since $A$ is commutative these are also ordinary modules over that ring. The action has quantum degree $- 2$ .

The square condition is that the sum of the two signed saddle composites around each two-dimensional cube face is zero. An edge sign function satisfying it is called square-compatible; existence and invariance require separate proofs.

Let $R = Z [h, X] / (X^{2} - h X)$ , with $\deg_{q} (h) = \deg_{q} (X) = - 2$ . Its Frobenius structure has unit $η (1) = 1$ , counit $ϵ (1) = 0, ϵ (X) = 1$ , quotient-ring multiplication, and comultiplication $Δ (1) = 1 \otimes X + X \otimes 1 - h 1 \otimes 1, Δ (X) = X \otimes X .$ Applying these operations to the same normalized cube gives the Bar–Natan cochain complex $C_{h} (D)$ , with circle labels $1, X$ of quantum degrees $1, - 1$ and the same normalized state shifts; it is a complex of free $R$ -modules, and the marked circle acts by multiplication by $X$ .

Put $B = F_{2} [X] / (X^{2}) = R / (2, h)$ . Modulo $2$ , the odd exterior basis vector $C_{i_{1}} \land \dots \land C_{i_{r}}$ corresponds to the even tensor state labeling the selected circles $C_{i_{j}}$ by $X$ and all other circles by $1$ . This statewise bijection defines the candidate comparison $f_{D} : C_{o} (D) \otimes F_{2} \to C_{h} (D) \otimes_{R} B$ after setting $h = 0$ .

A local even–odd (LEO) triple is a triple $(C, D, f)$ consisting of a finitely generated bigraded cochain complex $C$ free over $A$ , a finitely generated bigraded cochain complex $D$ free over $R$ , and a bigraded $F_{2} [X] / (X^{2})$ -linear chain homotopy equivalence $f : C \otimes_{Z} F_{2} ⟶ D \otimes_{R} B, B = R / (2, h) = F_{2} [X] / (X^{2}),$ such that $D [h^{- 1}]$ is chain homotopy equivalent over $R [h^{- 1}]$ to one free rank-one module supported in cohomological degree $0$ and odd quantum degree. For a based knot diagram the intended candidate is $(C_{o} (D), C_{h} (D), f_{D})$ , where $f_{D}$ identifies the reduced edge maps modulo $2$ and $h$ ; its satisfaction of the homotopy and localization conditions is a separate theorem.

A local map $(C, D, f) \to (C^{'}, D^{'}, f^{'})$ is a pair of bigraded module chain maps $α : C \to C^{'}$ and $β : D \to D^{'}$ such that $β [h^{- 1}]$ is a chain homotopy equivalence and the square induced by $f, f^{'}, α, β$ after quotienting by $(2, h)$ commutes up to an $F_{2} [X] / (X^{2})$ -linear chain homotopy. Two LEO triples are locally equivalent when there is a local map in each direction.

Definition references

Ozsváth–Rasmussen–Szabó, Odd Khovanov homology (2007), §§1.1–1.3

Bar-Natan, Khovanov's homology for tangles and cobordisms (2004), §9.3

Putyra, A 2-category of chronological cobordisms and odd Khovanov homology (2013), §§2–3

Dunfield–Lipshitz–Schütz, Local equivalence and refinements of Rasmussen's s-invariant (2023), §§2.1–3.1

Facts & Assumptions

Scope note: Bar-Natan §9.3 treats a characteristic-two tangle theory; the integral $Z [h]$ deformation defined above is given by the displayed formulas and is not attributed to that section.

Given: A finite regular based oriented link diagram and a square-compatible sign assignment.

[A1]

The sign data in the given odd cube are square-compatible, as stipulated in the Definition.

Proof

Given: The rings, state modules, saddle maps and quotient maps in the Definition.

1.1givenalgebra

The monic relations make $A$ free over $Z$ and $R$ free over $Z [h]$ , each with basis $1, X$ . Both relations are homogeneous for the displayed degrees. For the odd maps, the merge quotient is well-defined because $p (C_{1} - C_{2}) = 0$ . The split formula is independent of whether $j (C) = C_{1}$ or $C_{2}$ , since $(C_{1} - C_{2}) \land C_{1} = (C_{1} - C_{2}) \land C_{2} = C_{1} \land C_{2}$ .

2.1givenstep 1.1algebra

In $R$ , associativity, commutativity and the unit law follow from quotient-ring multiplication. On the basis $1, X$ , the counit composites $(ϵ \otimes i d) Δ$ and $(i d \otimes ϵ) Δ$ both send $1 \mapsto 1$ and $X \mapsto X$ . Cocommutativity follows from the symmetric formulas for $Δ (1)$ and $Δ (X)$ . For coassociativity, both iterated comultiplications send $X$ to $X \otimes X \otimes X$ and send $1$ to $1 \otimes X \otimes X + X \otimes 1 \otimes X + X \otimes X \otimes 1 - h (1 \otimes 1 \otimes X + 1 \otimes X \otimes 1 + X \otimes 1 \otimes 1) + h^{2} 1 \otimes 1 \otimes 1.$ Thus the unit, counit and coassociativity axioms hold.

3.1A1givenstep 1.1step 2.1algebra∎

On the ordered basis $1 \otimes 1, 1 \otimes X, X \otimes 1, X \otimes X$ , each of $Δ m$ , $(m \otimes i d) (i d \otimes Δ)$ , and $(i d \otimes m) (Δ \otimes i d)$ has respective outputs $1 \otimes X + X \otimes 1 - h 1 \otimes 1, X \otimes X, X \otimes X, h X \otimes X .$ Therefore the Frobenius identities hold. On every cube square, the two unsigned composites agree by tensor functoriality on disjoint circles, associativity for two merges, coassociativity for two splits, or the checked Frobenius identities for a merge-split pair. Opposite cube signs make the two signed composites cancel, so the Bar–Natan differential squares to zero. A based odd cube with the square-compatible sign data in [A1] is also a cochain complex: the coefficient of each height-two state in $d^{2}$ is the sum of its two signed composites, zero by the defining condition. The right marked action commutes with merges because the quotient carries $Z_{p}$ to the marked output circle. It commutes with splits because $((C_{1} - C_{2}) \land j (z)) \land Z_{p} = (C_{1} - C_{2}) \land j (z \land Z_{p})$ , including a split of the marked circle: either lift of the marked output differs by $C_{1} - C_{2}$ , whose extra wedge vanishes. Also $X^{2}$ acts by zero. Every state module is free over $A$ : choose the exterior products of unmarked circle generators as module basis and append $Z_{p}$ on the right for their $X$ multiples. Modulo $2$ this action is precisely the even marked action, and reduction of $R$ modulo $(2, h)$ retains that action through $B$ , rather than setting $X = 0$ . Existence of such odd sign assignments, the mod-2 comparison as a homotopy equivalence, and the localized rank-one condition are not claimed by this Definition; they are separate assertions of the foundations lemma.

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Sources

Peter Ozsváth, Jacob Rasmussen, and Zoltán Szabó, Odd Khovanov homology, arXiv:0710.4300 (standard reference, not scraped)
Dror Bar-Natan, Khovanov's homology for tangles and cobordisms, arXiv:math/0410495v2 (standard reference, not scraped)
Krzysztof K. Putyra, A 2-category of chronological cobordisms and odd Khovanov homology, arXiv:1310.1895 (standard reference, not scraped)
Nathan Dunfield, Robert Lipshitz, and Dirk Schütz, Local equivalence and refinements of Rasmussen's s-invariant, arXiv:2312.09114v3 (standard reference, not scraped)