Local equivalence and refinements of Rasmussen's s-invariant

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 C to the resulting local equivalence group CLEO of such triples. We give several versions of the s-invariant that descend to CLEO, including one that completely determines whether the image of a knot K in CLEO is trivial. We discuss computer experiments illustrating the power ofthese 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.