sl3–foam homology calculations

Lewark, Lukas¹

¹Institut de Mathématiques de Jussieu (IMJ) – Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3661-3686

Résumé

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots, which are counterexamples to Lobb’s conjecture that the sl3–knot concordance invariant s3 (suitably normalised) should be equal to the Rasmussen invariant s2. For this family, |s3| < |s2|. However, we also find other knots for which |s3| > |s2|. The main tool is an implementation of Morrison and Nieh’s algorithm to calculate Khovanov’s sl3–foam link homology. Our C++ program is fast enough to calculate the integral homology of, eg, the (6,5)–torus knot in six minutes. Furthermore, we propose a potential improvement of the algorithm by gluing sub-tangles in a more flexible way.

Détail

DOI : 10.2140/agt.2013.13.3661

Classification : 57M25, 81R50
Keywords: webs, foams, pretzel knots, four-ball genus, Khovanov–Rozansky homologies, Rasmussen invariant, $\mathfrak{sl}_N$ concordance invariants

Bibliographie

[1] S Akbulut, Cappell–Shaneson homotopy spheres are standard, Ann. of Math. 171 (2010) 2171

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

[3] D Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243

[4] C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883

[5] N Carqueville, D Murfet, Computing Khovanov–Rozansky homology and defect fusion

[6] A Champanerkar, I Kofman, Twisting quasi-alternating links, Proc. Amer. Math. Soc. 137 (2009) 2451

[7] M Freedman, Complexity classes as mathematical axioms, Ann. of Math. 170 (2009) 995

[8] M Freedman, R Gompf, S Morrison, K Walker, Man and machine thinking about the smooth $4$–dimensional Poincaré conjecture, Quantum Topol. 1 (2010) 171

[9] C M Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53

[10] J Green, S Morrison, JavaKh (2005)

[11] J Greene, Homologically thin, non-quasi-alternating links, Math. Res. Lett. 17 (2010) 39

[12] M Hedden, P Ording, The Ozsváth–Szabó and Rasmussen concordance invariants are not equal, Amer. J. Math. 130 (2008) 441

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

[14] F Jaeger, A new invariant of plane bipartite cubic graphs, Discrete Math. 101 (1992) 149

[15] T C Jaeger, Khovanov–Rozansky homology and Conway mutation (2011)

[16] V F R Jones, Planar algebras, I (1998)

[17] T Kawamura, The Rasmussen invariants and the sharper slice-Bennequin inequality on knots, Topology 46 (2007) 29

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

[19] M Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665

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

[21] M Khovanov, $\mathrm{sl}(3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045

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

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

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

[25] G Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996) 109

[26] L Lewark, FoamHo (2012)

[27] L Lewark, Rasmussen's spectral sequences and the $\mathfrak{sl}_n$–concordance invariants, in preparation (2013)

[28] C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735

[29] A Lobb, A slice genus lower bound from $\mathrm{sl}(n)$ Khovanov–Rozansky homology, Adv. Math. 222 (2009) 1220

[30] A Lobb, Computable bounds for Rasmussen's concordance invariant, Compos. Math. 147 (2011) 661

[31] A Lobb, A note on Gornik's perturbation of Khovanov–Rozansky homology, Algebr. Geom. Topol. 12 (2012) 293

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

[33] M Mackaay, P Vaz, The reduced HOMFLY-PT homology for the Conway and the Kinoshita–Terasaka knots (2008)

[34] C Manolescu, P Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links, from: "Proceedings of Gökova Geometry-Topology Conference 2007" (editors S Akbulut, T Önder, R J Stern), GGT Conference (2008) 60

[35] S Morrison, A Nieh, On Khovanov's cobordism theory for $\mathfrak{su}_3$ knot homology, J. Knot Theory Ramifications 17 (2008) 1121

[36] , MPIR, version 11.0.0 (2012)

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

[38] , PARI/GP, version 2.5.3 (2012)

[39] J A Rasmussen, Some differentials on Khovanov–Rozansky homology (2006)

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

[41] N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1

[42] A Shumakovitch, KhoHo (2003)

[43] A Shumakovitch, Torsion of the Khovanov homology (2004)

[44] A Shumakovitch, Khovanov homology theories and their applications, from: "Perspectives in analysis, geometry, and topology" (editors I Itenberg, B Jöricke, M Passare), Progr. Math. 296, Springer (2012) 403

[45] A Stoimenow, Knot data tables (2012)

[46] B Webster, Kr.m2 (2005)

[47] B Webster, Khovanov–Rozansky homology via a canopolis formalism, Algebr. Geom. Topol. 7 (2007) 673

[48] H Wu, The Khovanov–Rozansky cohomology and Bennequin inequalities (2007)