Deformations of colored 𝔰𝔩N link homologies via foams

Rose, David¹ ; Wedrich, Paul²

¹ University of North Carolina at Chapel Hill, Mathematics Department, 120 E Cameron Avenue, CB #3250, 329 Phillips Hall, Chapel Hill, NC 27599, United States
² Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom

Geometry & topology, Tome 20 (2016) no. 6, pp. 3431-3517.

Résumé

We prove a conjectured decomposition of deformed ${\mathfrak{s}\mathfrak{l}}_N$ link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored ${\mathfrak{s}\mathfrak{l}}_N$ link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.

Détail

DOI : 10.2140/gt.2016.20.3431

Classification : 17B37, 57M25, 81R50
Keywords: categorification, link homology, spectral sequence

Bibliographie

[1] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 | DOI

[2] D Bar-Natan, S Morrison, The Karoubi envelope and Lee’s degeneration of Khovanov homology, preprint (2009)

[3] H Becker, Khovanov–Rozansky homology via Cohen–Macaulay approximations and Soergel bimodules, preprint (2011)

[4] N Carqueville, D Murfet, Computing Khovanov–Rozansky homology and defect fusion, Algebr. Geom. Topol. 14 (2014) 489 | DOI

[5] N Carqueville, D Murfet, Adjunctions and defects in Landau–Ginzburg models, Adv. Math. 289 (2016) 480 | DOI

[6] S Cautis, Clasp technology to knot homology via the affine Grassmannian, Math. Ann. 363 (2015) 1053 | DOI

[7] S Cautis, J Kamnitzer, A Licata, Categorical geometric skew Howe duality, Invent. Math. 180 (2010) 111 | DOI

[8] S Cautis, J Kamnitzer, S Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014) 351 | DOI

[9] J Chuang, R Rouquier, Derived equivalences for symmetric groups and sl2–categorification, Ann. of Math. 167 (2008) 245 | DOI

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

[11] T Dyckerhoff, D Murfet, Pushing forward matrix factorizations, Duke Math. J. 162 (2013) 1249 | DOI

[12] B Gornik, Note on Khovanov link cohomology, preprint (2004)

[13] E Gorsky, S Gukov, M Stošić, Quadruply-graded colored homology of knots, preprint (2013)

[14] S Gukov, M Stošić, Homological algebra of knots and BPS states, from: "Proceedings of the Freedman Fest" (editors R Kirby, V Krushkal, Z Wang), Geom. Topol. Monogr. 18 (2012) 309 | DOI

[15] S Gukov, J Walcher, Matrix factorizations and Kauffman homology, preprint (2005)

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

[17] M Khovanov, Link homology and Frobenius extensions, Fund. Math. 190 (2006) 179 | DOI

[18] M Khovanov, A D Lauda, A diagrammatic approach to categorification of quantum groups, I, Represent. Theory 13 (2009) 309 | DOI

[19] M Khovanov, A D Lauda, A categorification of quantum sl(n), Quantum Topol. 1 (2010) 1 | DOI

[20] M Khovanov, A D Lauda, A diagrammatic approach to categorification of quantum groups, II, Trans. Amer. Math. Soc. 363 (2011) 2685 | DOI

[21] M Khovanov, A D Lauda, M Mackaay, M Stošić, Extended graphical calculus for categorified quantum sl(2), 1029, Amer. Math. Soc. (2012) | DOI

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

[23] D Krasner, Equivariant sl(n)–link homology, Algebr. Geom. Topol. 10 (2010) 1 | DOI

[24] A D Lauda, A categorification of quantum sl(2), Adv. Math. 225 (2010) 3327 | DOI

[25] A D Lauda, An introduction to diagrammatic algebra and categorified quantum sl2, Bull. Inst. Math. Acad. Sin. 7 (2012) 165

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

[27] L Lewark, Rasmussen’s spectral sequences and the slN–concordance invariants, Adv. Math. 260 (2014) 59 | DOI

[28] A Lobb, A slice genus lower bound from sl(n) Khovanov–Rozansky homology, Adv. Math. 222 (2009) 1220 | DOI

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

[30] G Lusztig, Introduction to quantum groups, 110, Birkhäuser (1993) | DOI

[31] I G Macdonald, Symmetric functions and Hall polynomials, Clarendon (1995)

[32] M Mackaay, The slN–web algebras and dual canonical bases, J. Algebra 409 (2014) 54 | DOI

[33] M Mackaay, W Pan, D Tubbenhauer, The sl3–web algebra, Math. Z. 277 (2014) 401 | DOI

[34] M Mackaay, P Vaz, The universal sl3–link homology, Algebr. Geom. Topol. 7 (2007) 1135 | DOI

[35] M Mackaay, Y Yonezawa, sl(N)–web categories, preprint (2013)

[36] H Queffelec, D E V Rose, The sln foam 2–category : a combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality, preprint (2014)

[37] J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419 | DOI

[38] J Rasmussen, Some differentials on Khovanov–Rozansky homology, Geom. Topol. 19 (2015) 3031 | DOI

[39] R Rouquier, 2–Kac–Moody algebras, preprint (2008)

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

[41] H Wu, Generic deformations of the colored sl(N)–homology for links, Algebr. Geom. Topol. 11 (2011) 2037 | DOI

[42] H Wu, Equivariant colored sl(N)–homology for links, J. Knot Theory Ramifications 21 (2012) | DOI

[43] H Wu, A colored sl(N) homology for links in S3, Dissertationes Math. (Rozprawy Mat.) 499 (2014) 217 | DOI

[44] Y Yonezawa, Quantum (sln,∧V n) link invariant and matrix factorizations, Nagoya Math. J. 204 (2011) 69