A note on Khovanov–Rozansky sl2-homology and ordinary Khovanov homology

Abstract

In this paper we present an explicit isomorphism between Khovanov–Rozansky sl2-homology and ordinary Khovanov homology. This result was originally claimed in Khovanov and Rozansky's paper [Matrix factorizations and link homology, Fund. Math.199(1) (2008) 1–91, MR 2391017 (2010a:57011)], though the proof was never presented. The main missing detail is providing a coherent choice of signs when identifying variables in the sl2-homology. Along with the behavior of the signs and local orientations in the sl2-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the sl2-homology may instead correspond to a different variant of Khovanov homology. In this paper we describe both theories and prove that they are in fact isomorphic by showing that a coherent choice of signs can be made. In doing so we emphasize the interpretation of the sl2-complex as a cube of resolutions.