Geometry & Topology

Some differentials on Khovanov–Rozansky homology

Jacob Rasmussen

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


We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N > 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR–homology of knots with 9 crossings or fewer.

Article information

Geom. Topol., Volume 19, Number 6 (2015), 3031-3104.

Received: 13 September 2006
Accepted: 21 January 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds

HOMFLY-PT categorification Khovanov–Rozansky differentials


Rasmussen, Jacob. Some differentials on Khovanov–Rozansky homology. Geom. Topol. 19 (2015), no. 6, 3031--3104. doi:10.2140/gt.2015.19.3031.

Export citation


  • 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
  • N M Dunfield, S Gukov, J Rasmussen, The superpolynomial for knot homologies, Experiment. Math. 15 (2006) 129–159
  • B Gornik, Note on Khovanov link cohomology
  • S Gukov, A Schwarz, C Vafa, Khovanov–Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53–74
  • J Hoste, M Thistlethwaite, Knotscape (1999) computer program. Available at \setbox0\makeatletter\@url {\unhbox0
  • V F R Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989) 311–334
  • L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407
  • M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
  • M Khovanov, Patterns in knot cohomology, I, Experiment. Math. 12 (2003) 365–374
  • 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, Fund. Math. 199 (2008) 1–91
  • M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387–1425
  • S Lang, Algebra, 3rd edition, Graduate Texts in Mathematics 211, Springer, New York (2002)
  • E S Lee, The support of the Khovanov's invariants for alternating knots
  • E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554–586
  • M Mackaay, P Vaz, The universal ${\rm sl}\sb 3$–link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169
  • J McCleary, User's guide to spectral sequences, Mathematics Lecture Series 12, Publish or Perish, Wilmington, DE (1985)
  • H Murakami, T Ohtsuki, S Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998) 325–360
  • P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225–254
  • J Rasmussen, Khovanov–Rozansky homology of two-bridge knots and links, Duke Math. J. 136 (2007) 551–583
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990)
  • V Turaev, Torsion invariants of ${\rm Spin}\sp c$–structures on $3$–manifolds, Math. Res. Lett. 4 (1997) 679–695
  • B Webster, KR.m2 (2005) Macaulay2 program. Available at \setbox0\makeatletter\@url {\unhbox0
  • H Wu, Braids, transversal links and the Khovanov–Rozansky theory, Trans. Amer. Math. Soc. 360 (2008) 3365–3389