Geometry & Topology

A geometric construction of colored HOMFLYPT homology

Benjamin Webster and Geordie Williamson

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Abstract

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.

Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 2557-2600.

Dates
Received: 28 September 2010
Revised: 25 June 2016
Accepted: 25 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859271

Digital Object Identifier
doi:10.2140/gt.2017.21.2557

Mathematical Reviews number (MathSciNet)
MR3687104

Zentralblatt MATH identifier
06774930

Subjects
Primary: 17B10: Representations, algebraic theory (weights) 57T10: Homology and cohomology of Lie groups

Keywords
knot homology triply graded homology

Citation

Webster, Benjamin; Williamson, Geordie. A geometric construction of colored HOMFLYPT homology. Geom. Topol. 21 (2017), no. 5, 2557--2600. doi:10.2140/gt.2017.21.2557. https://projecteuclid.org/euclid.gt/1510859271


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