A note on Khovanov-Rozansky sl(2)-homology and ordinary Khovanov homology

作者：Hughes Mark C^*

来源：Journal of Knot Theory and Its Ramifications, 2014, 23(12): 1450057.

DOI：10.1142/S0218216514500576

• Khovanov homology
• Khovanov-Rozansky homology
• Jones polynomial
• matrix factorizations

In this paper we present an explicit isomorphism between Khovanov-Rozansky sl(2)-homology and ordinary Khovanov homology. This result was originally claimed in Khovanov and Rozansky%26apos;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 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.

• 出版日期2014-10