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A geometric construction of colored HOMFLYPT homology

Ben Webster and Geordie Williamson

Geometry & Topology 21 (2017) 2557–2600

The aim of this paper is twofold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov and Rozansky. 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 it coincides with the categorification proposed by Mackaay, Stošić and Vaz.

knot homology, triply graded homology
Mathematical Subject Classification 2010
Primary: 17B10, 57T10
Received: 28 September 2010
Revised: 25 June 2016
Accepted: 25 December 2016
Published: 15 August 2017
Proposed: Peter S. Ozsváth
Seconded: Ciprian Manolescu, Haynes Miller
Ben Webster
Department of Pure Mathematics
University of Waterloo
and Perimeter Institute for Theoretical Physics
Waterloo ON
Geordie Williamson
School of Mathematics and Statistics
University of Sydney
Sydney NSW