Algebraic & Geometric Topology

Khovanov–Rozansky homology via a canopolis formalism

Ben Webster

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In this paper, we describe a canopolis (ie categorified planar algebra) formalism for Khovanov and Rozansky’s link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Khovanov–Rozansky homology and the definition using Soergel bimodules in a more general context, allow us to give a new proof of the invariance of triply graded homology and give a new analysis of the behavior of triply graded homology under the Reidemeister IIb move.

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 673-699.

Received: 23 February 2007
Accepted: 5 March 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 13D02: Syzygies, resolutions, complexes

Khovanov–Rozansky homology knot homology canopolis planar algebra


Webster, Ben. Khovanov–Rozansky homology via a canopolis formalism. Algebr. Geom. Topol. 7 (2007), no. 2, 673--699. doi:10.2140/agt.2007.7.673.

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  • D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370
  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press, Princeton, N.J. (1974)
  • D Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35–64
  • D Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer, New York (1995) With a view toward algebraic geometry
  • I Frenkel, M Khovanov, C Stroppel, A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products
  • V F R Jones, Planar algebras, I
  • M Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules
  • M Khovanov, L Rozansky, Matrix factorizations and link homology
  • M Khovanov, L Rozansky, Matrix factorizations and link homology II
  • J Rasmussen, Some differentials on Khovanov-Rozansky homology
  • R Rouquier, Categorification of the braid groups
  • W Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992) 49–74
  • C Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
  • C Stroppel, TQFT with corners and tilting functors in the Kac-Moody case