A note on Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology

Abstract

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of signs when identifying variables in the $sl_2$-homology. Along with the behavior of the signs and local orientations in the $sl_2$-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the $sl_2$-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 $sl_2$-complex as a cube of resolutions.