Theoretical analysis demonstrates a representation-theoretic construction of odd Khovanov homology, highlighting how algebraic signs emerge naturally from super-2-categorical structures.
We introduce a super analogue of gl₂ -foams and use it to define an invariant of oriented tangles, shown to coincide with odd Khovanov homology when restricted to links. We then define a supercategorification of the q -Schur algebra of level 2 and realize our construction as a certain super-2-representation. This gives a representation-theoretic construction of odd Khovanov homology, where signs naturally arise as a byproduct of the super-2-categorical structure. In the process, we define a tensor product for chain complexes in super-2-categories, suitably compatible with homotopies. This could be of independent interest.
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Schelstraete et al. (2026) studied this question.
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