Some differentials on Khovanov–Rozansky homology

Rasmussen, Jacob¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

Geometry & topology, Tome 19 (2015) no. 6, pp. 3031-3104.

Résumé

We study the relationship between the HOMFLY and $\mathit{sl}\left(N\right)$ 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 $\mathit{sl}\left(N\right)$ homology. As an application, we determine the KR–homology of knots with 9 crossings or fewer.

Détail

DOI : 10.2140/gt.2015.19.3031

Keywords: HOMFLY-PT, categorification, Khovanov–Rozansky, differentials

Bibliographie

[1] D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337

[2] D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443

[3] N M Dunfield, S Gukov, J Rasmussen, The superpolynomial for knot homologies, Experiment. Math. 15 (2006) 129

[4] B Gornik, Note on Khovanov link cohomology,

[5] S Gukov, A Schwarz, C Vafa, Khovanov–Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53

[6] J Hoste, M Thistlethwaite, Knotscape (1999)

[7] V F R Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989) 311

[8] L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395

[9] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[10] M Khovanov, Patterns in knot cohomology, I, Experiment. Math. 12 (2003) 365

[11] M Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Internat. J. Math. 18 (2007) 869

[12] M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1

[13] M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387

[14] S Lang, Algebra, Graduate Texts in Mathematics 211, Springer (2002)

[15] E S Lee, The support of the Khovanov's invariants for alternating knots,

[16] E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554

[17] M Mackaay, P Vaz, The universal $\mathrm{sl}_3$–link homology, Algebr. Geom. Topol. 7 (2007) 1135

[18] J Mccleary, User's guide to spectral sequences, Mathematics Lecture Series 12, Publish or Perish (1985)

[19] H Murakami, T Ohtsuki, S Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998) 325

[20] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225

[21] J Rasmussen, Khovanov–Rozansky homology of two-bridge knots and links, Duke Math. J. 136 (2007) 551

[22] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990)

[23] V Turaev, Torsion invariants of $\mathrm{Spin}^c$–structures on $3$–manifolds, Math. Res. Lett. 4 (1997) 679

[24] B Webster, KR.m2 (2005)

[25] H Wu, Braids, transversal links and the Khovanov–Rozansky theory, Trans. Amer. Math. Soc. 360 (2008) 3365