Not even Khovanov homology

TL;DR

This paper introduces a new supercategory-based homology theory for links that supercategorifies the Jones polynomial, distinct from even Khovanov homology and potentially isomorphic to odd Khovanov homology.

Contribution

It constructs a supercategory as a skew version of KLR algebras and develops a link homology theory that supercategorifies the Jones polynomial, with applications to representations of rak{gl}_n.

Findings

The homology theory is distinct from even Khovanov homology.

Evidence suggests the homology may be isomorphic to odd Khovanov homology.

Cyclotomic quotients categorify irreducible rak{gl}_n} representations at level 2.

Abstract

We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology theory supercategorifying the Jones polynomial. Our homology is distinct from even Khovanov homology and we present evidence supporting the conjecture that it is isomorphic to odd Khovanov homology. We also show that cyclotomic quotients of our supercategory give supercategorifications of irreducible finite-dimensional representations of $\mathfrak{gl}_n$ of level 2.