On Khovanov local equivalence groups

Abstract

Inspired by work in Seiberg--Witten and Heegaard Floer homology, Dunfield, Schütz, and the author constructed an abelian "local equivalence group" built from pairs of chain complexes, and a homomorphism from the smooth concordance group to this group that uses the odd Khovanov complex and the Bar-Natan deformation of the even Khovanov complex. Lewark independently gave a related local equivalence group and concordance invariant, using only the Bar-Natan deformation. In this paper, we construct $(\mathbb{Z}/2\mathbb{Z})^\infty$ subgroups of Lewark's local equivalence group and the kernel of the map from our local equivalence group to Lewark's, as well as $\mathbb{Z}^\infty$ direct summands of Lewark's group and this kernel, invisible to the $s$-invariants. The construction gives at least six $\mathbb{Z}$-valued concordance homomorphisms linearly independent from $s$, $τ$, and Dai--Hom--Stoffregen--Truong's invariants.