A geometric construction of colored HOMFLYPT homology
Abstract
The aim of this paper is twofold. First, we give a fully geometric description of the HOMFLYPT homology of KhovanovRozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology. We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that this coincides with the categorification proposed by Mackaay, Stosic and Vaz.
 Publication:

arXiv eprints
 Pub Date:
 May 2009
 arXiv:
 arXiv:0905.0486
 Bibcode:
 2009arXiv0905.0486W
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Geometry;
 17B10;
 57T10
 EPrint:
 31 pages