Symmetric Khovanov-Rozansky link homologies

Louis-Hadrien Robert; Emmanuel Wagner

doi:10.5802/jep.124

Symmetric Khovanov-Rozansky link homologies
[Homologies d’entrelacs de Khovanov–Rozansky symétriques]

Louis-Hadrien Robert ¹ ; Emmanuel Wagner ²

¹ Université de Genève 2–4 rue du lièvre, 1227 Genève, Switzerland Cogitamus Laboratory
² Univ Paris Diderot, IMJ-PRG, UMR 7586 CNRS F-75013, Paris, France Université de Bourgogne Franche-Comté, IMB, UMR 5584 21000 Dijon, France Cogitamus Laboratory

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 573-651.

Abstract (VO)
Résumé

We provide a finite-dimensional categorification of the symmetric evaluation of ${𝔰𝔩}_{N}$ -webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of ${𝔰𝔩}_{N}$ . The construction is made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

On donne une catégorification de l’évaluation symétrique des toiles ${𝔰𝔩}_{N}$ en utilisant les mousses. On en déduit des théories homologiques d’entrelacs qui catégorifient les invariants quantiques d’entrelacs associés aux puissances symétriques de la représentation standard de ${𝔰𝔩}_{N}$ . Ces théories sont obtenues dans un cadre équivariant. On montre qu’il existe des suites spectrales de l’homologie triplement graduée de Khovanov-Rozansky vers ces homologies symétriques. On donne aussi une interpretation des bimodules de Soergel en terme de mousses.

Reçu le : 2018-07-13
Accepté le : 2020-03-16
Publié le : 2020-04-02

DOI : 10.5802/jep.124

Classification : 57R56, 57M27, 17B10, 17B37
Keywords: Link homology, quantum invariant, foams, Soergel bimodules
Mots-clés : Homologie d’entrelacs, invariant quantiques, mousses, bimodules de Soergel

Affiliations des auteurs :

Louis-Hadrien Robert ¹ ; Emmanuel Wagner ²

¹ Université de Genève 2–4 rue du lièvre, 1227 Genève, Switzerland Cogitamus Laboratory
² Univ Paris Diderot, IMJ-PRG, UMR 7586 CNRS F-75013, Paris, France Université de Bourgogne Franche-Comté, IMB, UMR 5584 21000 Dijon, France Cogitamus Laboratory

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Louis-Hadrien Robert and Emmanuel Wagner},
     title = {Symmetric {Khovanov-Rozansky} link homologies},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {573--651},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.124},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.124/}
}

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Louis-Hadrien Robert; Emmanuel Wagner. Symmetric Khovanov-Rozansky link homologies. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 573-651. doi : 10.5802/jep.124. https://jep.centre-mersenne.org/articles/10.5802/jep.124/

Bibliographie
Cité par

[BHMV95] C. Blanchet, N. Habegger, G. Masbaum & P. Vogel - “Topological quantum field theories derived from the Kauffman bracket”, Topology 34 (1995) no. 4, p. 883-927 | DOI | MR | Zbl

[Bla10] C. Blanchet - “An oriented model for Khovanov homology”, J. Knot Theory Ramifications 19 (2010) no. 2, p. 291-312 | DOI | MR | Zbl

[BLS18] R. Berger, T. Lambre & A. Solotar - “Koszul calculus”, Glasgow Math. J. 60 (2018) no. 2, p. 361-399, Addendum: Ibid. 61 (2019), no. 1, p. 249 | DOI | MR | Zbl

[BN05] D. Bar-Natan - “Khovanov’s homology for tangles and cobordisms”, Geom. Topol. 9 (2005), p. 1443-1499 | DOI | MR | Zbl

[Bro82] K. S. Brown - Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York-Berlin, 1982 | Zbl

[BW08] A. Beliakova & S. Wehrli - “Categorification of the colored Jones polynomial and Rasmussen invariant of links”, Canad. J. Math. 60 (2008) no. 6, p. 1240-1266 | DOI | MR | Zbl

[Cau15] S. Cautis - “Clasp technology to knot homology via the affine Grassmannian”, Math. Ann. 363 (2015) no. 3-4, p. 1053-1115 | DOI | MR | Zbl

[Cau17] S. Cautis - “Remarks on coloured triply graded link invariants”, Algebraic Geom. Topol. 17 (2017) no. 6, p. 3811-3836 | DOI | MR | Zbl

[CH15] B. Cooper & M. Hogancamp - “An exceptional collection for Khovanov homology”, Algebraic Geom. Topol. 15 (2015) no. 5, p. 2659-2707 | DOI | MR | Zbl

[CK08a] S. Cautis & J. Kamnitzer - “Knot homology via derived categories of coherent sheaves. I. The $𝔰𝔩 (2)$ -case”, Duke Math. J. 142 (2008) no. 3, p. 511-588 | DOI

[CK08b] S. Cautis & J. Kamnitzer - “Knot homology via derived categories of coherent sheaves. II: ${𝔰𝔩}_{m}$ case”, Invent. Math. 174 (2008) no. 1, p. 165-232 | DOI

[CK12] B. Cooper & V. Krushkal - “Categorification of the Jones-Wenzl projectors”, Quantum Topol. 3 (2012) no. 2, p. 139-180 | DOI | MR | Zbl

[CKW09] A. Conca, C. Krattenthaler & J. Watanabe - “Regular sequences of symmetric polynomials”, Rend. Sem. Mat. Univ. Padova 121 (2009), p. 179-199 | DOI | Numdam | MR | Zbl

[Dow17] N. Dowlin - “A categorification of the HOMFLY-PT polynomial with a spectral sequence to knot Floer homology”, 2017 | arXiv

[ES02] E. J. Elizondo & V. Srinivas - “Some remarks on Chow varieties and Euler-Chow series”, J. Pure Appl. Algebra 166 (2002) no. 1, p. 67 -81 | DOI | MR | Zbl

[EST17] M. Ehrig, C. Stroppel & D. Tubbenhauer - “The Blanchet-Khovanov algebras”, in Categorification and higher representation theory, Contemp. Math., vol. 683, American Mathematical Society, Providence, RI, 2017, p. 183-226 | DOI | MR | Zbl

[ETW18] M. Ehrig, D. Tubbenhauer & P. Wedrich - “Functoriality of colored link homologies”, Proc. London Math. Soc. (3) 117 (2018) no. 5, p. 996-1040 | DOI | MR | Zbl

[FSS12] I. Frenkel, C. Stroppel & J. Sussan - “Categorifying fractional Euler characteristics, Jones-Wenzl projectors and $3 j$ -symbols”, Quantum Topol. 3 (2012) no. 2, p. 181-253 | DOI | Zbl

[Ful07] W. Fulton - “Equivariant Cohomology in Algebraic Geometry”, Eilenberg lectures, 2007, https://people.math.osu.edu/anderson.2804/eilenberg/

[GGS18] E. Gorsky, S. Gukov & M. Stošić - “Quadruply-graded colored homology of knots”, Fund. Math. 243 (2018) no. 3, p. 209-299 | DOI | MR | Zbl

[Kas04] C. Kassel - “Homology and cohomology of associative algebras — A concise introduction to cyclic homology”, Notes of a course given in the Advanced School on Non-commutative Geometry at ICTP, 2004, 37 pages, http://www-irma.u-strasbg.fr/~kassel/Kassel-ICTPnotes2004.pdf

[Kau13] L. H. Kauffman - Knots and physics, World Scientific, Singapore, 2013 | Zbl

[Kho04] M. Khovanov - “sl(3) link homology”, Algebraic Geom. Topol. 4 (2004), p. 1045-1081 | DOI | MR | Zbl

[Kho05] M. Khovanov - “Categorifications of the colored Jones polynomial”, J. Knot Theory Ramifications 14 (2005) no. 1, p. 111-130 | DOI | MR | Zbl

[Kho07] M. Khovanov - “Triply-graded link homology and Hochschild homology of Soergel bimodules”, Internat. J. Math. 18 (2007) no. 8, p. 869-885 | DOI | MR | Zbl

[KR08] M. Khovanov & L. Rozansky - “Matrix factorizations and link homology. II”, Geom. Topol. 12 (2008) no. 3, p. 1387-1425 | DOI | MR | Zbl

[Kra10a] D. Krasner - “Equivariant $sl (n)$ -link homology”, Algebraic Geom. Topol. 10 (2010) no. 1, p. 1-32 | DOI | MR

[Kra10b] D. Krasner - “Integral HOMFLY-PT and $sl (n)$ -link homology”, Internat. J. Math. Math. Sci. (2010), article ID 896879, 25 pages | DOI | MR

[Lan02] S. Lang - Algebra, Graduate Texts in Math., vol. 211, Springer-Verlag, New York, 2002 | DOI | Zbl

[Lod98] J.-L. Loday - Cyclic homology, Grundlehren Math. Wiss., vol. 301, Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[LQR15] A. D. Lauda, H. Queffelec & D. E. V. Rose - “Khovanov homology is a skew Howe 2-representation of categorified quantum ${𝔰𝔩}_{m}$ ”, Algebraic Geom. Topol. 15 (2015) no. 5, p. 2517-2608 | DOI

[Lus10] G. Lusztig - Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010, Reprint of the 1994 edition | DOI | Zbl

[Mac15] I. G. Macdonald - Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015

[MOY98] H. Murakami, T. Ohtsuki & S. Yamada - “Homfly polynomial via an invariant of colored plane graphs”, Enseign. Math. (2) 44 (1998) no. 3-4, p. 325-360 | MR | Zbl

[MS09] V. Mazorchuk & C. Stroppel - “A combinatorial approach to functorial quantum ${𝔰𝔩}_{k}$ knot invariants”, Amer. J. Math. 131 (2009) no. 6, p. 1679-1713 | DOI

[MSV09] M. Mackaay, M. Stošić & P. Vaz - “ $𝔰𝔩 (N)$ -link homology $(N \geq 4)$ using foams and the Kapustin-Li formula”, Geom. Topol. 13 (2009) no. 2, p. 1075-1128 | DOI

[MSV11] M. Mackaay, M. Stošić & P. Vaz - “The $1, 2$ -coloured HOMFLY-PT link homology”, Trans. Amer. Math. Soc. 363 (2011) no. 4, p. 2091-2124 | DOI

[MW18] M. Mackaay & B. Webster - “Categorified skew Howe duality and comparison of knot homologies”, Adv. Math. 330 (2018), p. 876-945 | DOI | MR | Zbl

[QR16] H. Queffelec & D. E. V. Rose - “The ${𝔰𝔩}_{n}$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality”, Adv. Math. 302 (2016), p. 1251-1339 | DOI | MR

[QR18] H. Queffelec & D. E. V. Rose - “Sutured annular Khovanov-Rozansky homology”, Trans. Amer. Math. Soc. 370 (2018) no. 2, p. 1285-1319 | DOI | MR | Zbl

[QRS18] H. Queffelec, D. E. V. Rose & A. Sartori - “Annular evaluation and link homology”, 2018 | arXiv | Zbl

[Ras15] J. Rasmussen - “Some differentials on Khovanov-Rozansky homology”, Geom. Topol. 19 (2015) no. 6, p. 3031-3104 | DOI | MR | Zbl

[Rou17] R. Rouquier - “Khovanov-Rozansky homology and 2-braid groups”, in Categorification in geometry, topology, and physics, Contemp. Math., vol. 684, American Mathematical Society, Providence, RI, 2017, p. 147-157 | DOI | MR | Zbl

[Roz14] L. Rozansky - “An infinite torus braid yields a categorified Jones-Wenzl projector”, Fund. Math. 225 (2014) no. 1, p. 305-326 | DOI | MR | Zbl

[RT16] D. E. V. Rose & D. Tubbenhauer - “Symmetric webs, Jones-Wenzl recursions, and $q$ -Howe duality”, Internat. Math. Res. Notices 2016 (2016) no. 17, p. 5249-5290 | DOI

[RW16] D. E. V. Rose & P. Wedrich - “Deformations of colored ${𝔰𝔩}_{N}$ link homologies via foams”, Geom. Topol. 20 (2016) no. 6, p. 3431-3517 | DOI

[RW17] L.-H. Robert & E. Wagner - “A closed formula for the evaluation of ${𝔰𝔩}_{N}$ -foams”, 2017, To appear in Quantum Topology | arXiv

[Soe92] W. Soergel - “The combinatorics of Harish-Chandra bimodules”, J. reine angew. Math. 429 (1992), p. 49-74 | DOI | MR | Zbl

[SS14] C. Stroppel & J. Sussan - “Categorified Jones-Wenzl projectors: a comparison”, in Perspectives in representation theory, Contemp. Math., vol. 610, American Mathematical Society, Providence, RI, 2014, p. 333-351 | DOI | MR | Zbl

[ST19] A. Sartori & D. Tubbenhauer - “Webs and $q$ -Howe dualities in types BCD”, Trans. Amer. Math. Soc. 371 (2019) no. 10, p. 7387-7431 | DOI | Zbl

[Sto08] M. Stošić - “Hochschild homology of certain Soergel bimodules”, 2008 | arXiv | Zbl

[Str04] C. Stroppel - “A structure theorem for Harish-Chandra bimodules via coinvariants and Golod rings”, J. Algebra 282 (2004) no. 1, p. 349-367 | DOI | MR | Zbl

[Sus07] J. Sussan - Category O and sl(k) link invariants, Ph. D. Thesis, Yale University, Ann Arbor, MI, 2007 | arXiv | MR

[TVW17] D. Tubbenhauer, P. Vaz & P. Wedrich - “Super $q$ -Howe duality and web categories”, Algebraic Geom. Topol. 17 (2017) no. 6, p. 3703-3749 | DOI | Zbl

[Vaz08] P. Vaz - A categorification of the quantum sl(N)-link polynomials using foams, Ph. D. Thesis, Universidade do Algarve, 2008 | arXiv

[Web17] B. Webster - Knot invariants and higher representation theory, Mem. Amer. Math. Soc., vol. 250, no. 1191, American Mathematical Society, Providence, RI, 2017 | DOI | MR | Zbl

[Wed19] P. Wedrich - “Exponential growth of colored HOMFLY-PT homology”, Adv. Math. 353 (2019), p. 471-525 | DOI | MR | Zbl

[Wil11] G. Williamson - “Singular Soergel bimodules”, Internat. Math. Res. Notices 2011 (2011) no. 20, p. 4555-4632 | DOI | MR | Zbl

[Wu13] H. Wu - “Colored Morton-Franks-Williams inequalities”, Internat. Math. Res. Notices 2013 (2013) no. 20, p. 4734-4757 | DOI | MR | Zbl

[Wu14] H. Wu - “A colored $𝔰𝔩 (N)$ homology for links in $S^{3}$ ”, Dissertationes Math. (Rozprawy Mat.) 499 (2014), p. 1-217 | DOI

[WW17] B. Webster & G. Williamson - “A geometric construction of colored HOMFLYPT homology”, Geom. Topol. 21 (2017) no. 5, p. 2557-2600 | DOI | MR | Zbl

[Yon11] Y. Yonezawa - “Quantum $({𝔰𝔩}_{n}, \land V_{n})$ link invariant and matrix factorizations”, Nagoya Math. J. 204 (2011), p. 69-123 | DOI | MR

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