Local equivalence and refinements of Rasmussen's s‐invariant

Abstract Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even–odd (LEO) triple. We get a homomorphism from the smooth concordance group to the resulting local equivalence group of such triples. We give several versions of the ‐invariant that descend to , including one that completely determines whether the image of a knot in is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu–Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.