Geometry & Topology

A geometric construction of colored HOMFLYPT homology

Benjamin Webster and Geordie Williamson

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

knot homology triply graded homology


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.

Export citation


  • M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas, Tome 3: Exposés IX–XIX (SGA $4\ssty\sb{\,\mathrm{3}}\kern-.1em$), Lecture Notes in Math. 305, Springer (1973)
  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • A Beilinson, V Ginzburg, W Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996) 473–527
  • J Bernstein, V Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer (1994)
  • A A Beĭlinson, J Bernstein, P Deligne, Faisceaux pervers, from “Analysis and topology on singular spaces, I”, Astérisque 100, Soc. Math. France, Paris (1982) 5–171
  • M V Bondarko, Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey, preprint (2009)
  • M Brion, Equivariant cohomology and equivariant intersection theory, from “Representation theories and algebraic geometry” (A Broer, G Sabidussi, editors), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad., Dordrecht (1998) 1–37
  • P Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. (1971) 5–57
  • P Deligne, Cohomologie étale (SGA $4\chalf\kern-.1em$), Lecture Notes in Math. 569, Springer (1977)
  • P Deligne, La conjecture de Weil, II, Inst. Hautes Études Sci. Publ. Math. (1980) 137–252
  • V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335–388
  • M Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Internat. J. Math. 18 (2007) 869–885
  • M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387–1425
  • R Kiehl, R Weissauer, Weil conjectures, perverse sheaves and $l$–adic Fourier transform, Ergeb. Math. Grenzgeb. 42, Springer (2001)
  • G Laumon, Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. (1987) 131–210
  • X-S Lin, H Zheng, On the Hecke algebras and the colored HOMFLY polynomial, Trans. Amer. Math. Soc. 362 (2010) 1–18
  • M Mackaay, M Stošić, P Vaz, The $1,2$–coloured HOMFLY-PT link homology, Trans. Amer. Math. Soc. 363 (2011) 2091–2124
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Math. 58, Cambridge Univ. Press (2001)
  • H Murakami, T Ohtsuki, S Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998) 325–360
  • D Pauksztello, Compact corigid objects in triangulated categories and co–$t$–structures, Cent. Eur. J. Math. 6 (2008) 25–42
  • C A M Peters, J H M Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. 52, Springer (2008)
  • M Saito, Mixed Hodge modules, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) 360–363
  • C Stroppel, A structure theorem for Harish-Chandra bimodules via coinvariants and Golod rings, J. Algebra 282 (2004) 349–367
  • B Webster, Khovanov–Rozansky homology via a canopolis formalism, Algebr. Geom. Topol. 7 (2007) 673–699
  • B Webster, G Williamson, The bounded below equivariant derived category, in preparation
  • B Webster, G Williamson, A geometric model for Hochschild homology of Soergel bimodules, Geom. Topol. 12 (2008) 1243–1263
  • B Webster, G Williamson, The geometry of Markov traces, preprint (2009)
  • G Williamson, Singular Soergel bimodules, Int. Math. Res. Not. 2011 (2011) 4555–4632