A module structure on odd Khovanov homology and the odd invariant for ribbon 2-knots

Abstract: We prove that the reduced odd Khovanov homology of a link $L$ is naturally a module over the exterior algebra of the first homology of the link's branched double-cover. We then describe this module structure more geometrically and related it to the odd Khovanov maps induced by link cobordisms. As an application, we will give a combinatorial proof of a recent result of Spyropoulos-Vidyarthi-Zhang about the odd invariant for $2$-knots in the special case where the $2$-knot is a ribbon $2$-knot. Additionally, we will show that Levine-Zemke's main result from their 2019 paper on Khovanov homology and ribbon concordance remains true for odd Khovanov homology with rational coefficients and with coefficients in $\mathbb{Z}_{2<sup>k}$.

• The paper establishes a canonical module structure on reduced odd Khovanov homology over the exterior algebra of H1 of the branched double-cover.
• It provides explicit combinatorial formulas via dot maps and cobordism maps that link topological actions to ribbon 2-knot invariants.
• The work proves injectivity under ribbon concordances and explains p-torsion phenomena in odd Khovanov homology for pretzel knots.

A Module Structure on Odd Khovanov Homology and the Odd Invariant for Ribbon 2-Knots

Introduction and Motivation

This work establishes a canonical module structure on the reduced odd Khovanov homology of a link $L$ over the exterior algebra of the first homology group of the branched double-cover $\Sigma(L)$. The odd Khovanov homology, introduced by Ozsváth, Rasmussen, and Szabó, differs markedly from its even counterpart over $\mathbb{Z}$ or $\mathbb{Q}$ coefficients, and its connections to $\Sigma(L)$ arise in various contexts, such as spectral sequences converging to Floer theories, mutation invariance, and functorial constructions.

Central results include a combinatorial definition of the module action, explicit formulae for the action related to cobordism maps, and consequences for 2-dimensional knot invariants. Applications cover a combinatorial proof of a result by Spyropoulos-Vidyarthi-Zhang connecting the odd invariant for ribbon 2-knots to the order of the first homology group of their branched covers, as well as an odd Khovanov analog of Levine-Zemke's injectivity theorem for ribbon concordances. The module structure is geometrically described and shown to be compatible with the various flavors (types X and Y) of odd Khovanov homology.

Odd Khovanov Homology and Module Structure

Foundations

Odd Khovanov homology $OKh(L)$ is defined via a functorial construction involving chronological cobordisms, as formulated by Putyra, and a sign assignment (type X or Y) that governs the anti-/commutativity of differentials around certain 2-faces of the cube of resolutions. The reduced theory $\overline{OKh}(L)$ is obtained via a canonical subquotient corresponding to the reduced exterior algebra built from the link diagram, and when coefficients are in $\mathbb{F}_2$, it agrees with the even theory.

Main Theorem: Exterior Algebra Module Structure

The principal advance of the paper is the proof that $\overline{OKh}(L)$ is a module over the exterior algebra $\Lambda^*H_1(\Sigma(L);\mathbb{Z})$. Nonreduced odd Khovanov homology $\Sigma(L)$0 similarly carries an action by $\Sigma(L)$1, in line with long-standing expectations and analogies from even theory and Heegaard Floer homology.

The module action arises combinatorially via "dot maps"—chain maps associated to placements of dots on diagram edges, which under the functor to modules, become wedge multiplications by the associated generators of $\Sigma(L)$2. Relations among dot maps mirror coloring module relations for link diagrams, and when $\Sigma(L)$3 is homologically thin (e.g., alternating links), the module action is trivial for degree reasons.

Explicit Action, Types X and Y, and Cobordism Maps

The paper provides a precise geometric description: the module action of a class represented by a geometric arc $\Sigma(L)$4 (connecting boundary points on the diagram) is a specific signed sum of dot chain maps over the sequence of over- and under-crossings. For type X and Y odd Khovanov homology, the action formula references the specific arc (overcrossings for type X, undercrossings for type Y). Furthermore, the canonical isomorphism between type X and type Y theories intertwines the two module structures, yielding full naturality.

Cobordisms—properly embedded smooth surfaces in $\Sigma(L)$5—induce maps on $\Sigma(L)$6, and certain decorated cobordisms (with arcs or tubes attached) correspond precisely to module actions by homology classes, making the connection between geometric transformations and algebraic module structure transparent.

Applications and Numerical Results

Odd Invariant of Ribbon 2-Knots

Utilizing the module action, the work gives a combinatorial proof that the odd invariant $\Sigma(L)$7 for a ribbon 2-knot $\Sigma(L)$8 equals the order $\Sigma(L)$9, confirming a conjecture previously approached analytically by [SVZ2026]. The proof exploits the handle decomposition induced by a ribbon presentation and reduces the computation to a determinant of a linking matrix, shown to compute the order of the first homology.

Injection Theorems for Ribbon Concordances

In direct analogy with Levine-Zemke's theorem for the even theory, it is shown that any ribbon concordance induces an injective map on odd Khovanov homology, provided coefficients are taken in $\mathbb{Z}$0 or $\mathbb{Z}$1. Here, the key point is that the corresponding cobordism induces a multiplication by a nonvanishing (odd) integer on the homology, hence invertible over the specified coefficient rings.

Torsion and Nontrivial Module Actions

The construction elucidates previously mysterious occurrences of $\mathbb{Z}$2-torsion in odd Khovanov homology for certain pretzel knots (e.g., the $\mathbb{Z}$3-pretzel knot), demonstrating that module actions associated to specific homology classes realize nontrivial maps from the free part to the torsion part. This offers an explanation for the patterns noted by Shumakovitch and subsequently generalized in the literature.

Theoretical Implications and Functoriality

Extension of Functoriality and Decorated Cobordisms

The theory is framed categorically, defining a category of decorated link cobordisms, $\mathbb{Z}$4, where morphisms are pairs of a cobordism and a homology class element, with composition rules reflecting algebraic exterior product and grading (superdegree) interactions. Odd Khovanov homology extends to a functor (up to sign) from this category, encapsulating both module structure and cobordism maps in a unified framework.

Equivalence to Dotted Cobordism Category

A detailed construction of a dotted cobordism category, where cobordisms are equipped with dot configurations satisfying natural relations, is shown to be equivalent to the decorated algebraic category. This provides flexibility: topological moves and algebraic relations are translated through a combinatorial calculus, clarifying the connections between geometric and algebraic perspectives.

Limitations and Non-functoriality in $\mathbb{Z}$5

The work explicitly demonstrates, through computations involving constructed families of cobordisms, that odd Khovanov homology is not functorial for general ambient isotopy in $\mathbb{Z}$6. The structure is inherently sensitive to the embedding into $\mathbb{Z}$7 rather than $\mathbb{Z}$8, echoing phenomena already observed in even theory but with unique odd-specific manifestations.

Conclusion

This research rigorously establishes a natural module structure for odd Khovanov homology over the exterior algebra of the first homology of the branched double-cover, extends functorial constructions to a category of decorated (and dotted) cobordisms, and applies these structures to extract strong structural, numerical, and functorial consequences in low-dimensional topology. The connections between geometric, combinatorial, and algebraic data are made explicit, resolving several open conjectures and integrating odd Khovanov homology into the broader framework of algebraic structures reminiscent of Floer theoretic and quantum invariants.

These advances open new avenues for the study of categorified invariants of links and surfaces, their torsion phenomena, and their interactions with 4-manifold topology. The categorical and functorial setup also facilitates further extensions, such as actions by Lie superalgebras (cf. $\mathbb{Z}$9 in (Ebert et al., 2 Nov 2025)), and may provide templates for analogous constructions in related categorifications and homological invariants.