Comptes Rendus
Homological algebra/Topology
Perverse sheaves and knot contact homology
Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 378-399.

In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in R3, we define a differential graded (DG) k-category A˜L with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of a distinguished object of A˜L coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng, and Sullivan in [13]. The input of our construction is a natural action of the braid group Bn on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson, and Vilonen in [19]. As an application, we show that the category of finite-dimensional representations of the link k-category A˜L=H0(A˜L) defined as the 0-th homology of A˜L is equivalent to the category of perverse sheaves on R3 that are singular along the link L. We also obtain several generalizations of the category A˜L by extending the Gelfand–MacPherson–Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4].

Dans cette Note, nous donnons une nouvelle construction algébrique de l'homologie de contact des nœuds, au sens de Ng [37]. Pour un entrelacs L dans R3, nous définissons une k-catégorie différentielle graduée A˜L ayant un nombre fini d'objets, dont la classe de quasi-équivalence est un invariant topologique de L. Lorsque L est un nœud, l'algèbre des endomorphismes d'un objet distingué de A˜L coïncide avec l'algèbre différentielle graduée, pleinement non commutative du nœud, définie par Ekholm, Etnyre, Ng et Sullivan dans [12]. Notre construction se base sur une action naturelle du groupe de tresses Bn sur la catégorie des faisceaux pervers sur un disque de dimension deux avec singularités en n points marqués, étudiée par Gelfand, McPherson et Vilonen dans [19]. Comme application, nous montrons que la catégorie des représentations de dimension finie de la k-catégorie d'entrelacs A˜L=H0(A˜L), définie comme l'homologie de degré 0 de A˜L, est équivalente à la catégorie des faisceaux pervers sur R3 qui sont singuliers le long de l'entrelacs L. Nous obtenons également plusieurs généralisations de la catégorie A˜L en étendant l'action du groupe de tresses de Gelfand–MacPherson–Vilonen.

Published online:
DOI: 10.1016/j.crma.2017.02.007

Yuri Berest 1; Alimjon Eshmatov 2; Wai-Kit Yeung 1

1 Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA
2 Department of Mathematics and Statistics, University of Toledo, Toledo, OH 43606-3390, USA
     author = {Yuri Berest and Alimjon Eshmatov and Wai-Kit Yeung},
     title = {Perverse sheaves and knot contact homology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {378--399},
     publisher = {Elsevier},
     volume = {355},
     number = {4},
     year = {2017},
     doi = {10.1016/j.crma.2017.02.007},
     language = {en},
AU  - Yuri Berest
AU  - Alimjon Eshmatov
AU  - Wai-Kit Yeung
TI  - Perverse sheaves and knot contact homology
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 378
EP  - 399
VL  - 355
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2017.02.007
LA  - en
ID  - CRMATH_2017__355_4_378_0
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%A Yuri Berest
%A Alimjon Eshmatov
%A Wai-Kit Yeung
%T Perverse sheaves and knot contact homology
%J Comptes Rendus. Mathématique
%D 2017
%P 378-399
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Yuri Berest; Alimjon Eshmatov; Wai-Kit Yeung. Perverse sheaves and knot contact homology. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 378-399. doi : 10.1016/j.crma.2017.02.007.

[1] R. Anno; T. Logvinenko Spherical DG functors | arXiv

[2] E. Artin Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb., Volume 4 (1925), pp. 47-72

[3] H.J. Baues; J.-M. Lemaire Minimal models in homotopy theory, Math. Ann., Volume 225 (1977), pp. 219-242

[4] Yu. Berest, A. Eshmatov, W.K. Yeung, Homotopy braid closure and link invariants, in preparation.

[5] Yu. Berest, A.C. Ramadoss, W.K. Yeung, Representation homology of spaces and higher Hochschild homology, in preparation.

[6] J.S. Birman Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, vol. 82, Princeton University Press/University of Tokyo Press, Princeton, NJ/Tokyo, 1974

[7] J.S. Carter; D.S. Silver; S.G. Williams; M. Elhamdadi; M. Saito Virtual knot invariants from group biquandles and their cocycles, J. Knot Theory Ramif., Volume 18 (2009), pp. 957-972

[8] J. Crisp; L. Paris Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, Pac. J. Math., Volume 221 (2005), pp. 1-27

[9] W.G. Dwyer; J. Spalinski, Elsevier (1995), pp. 73-126

[10] W. Dwyer; P. Hirschhorn; D. Kan; J. Smith Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs, vol. 113, AMS, Providence, RI, 2004

[11] T. Ekholm; J. Etnyre; M. Sullivan The contact homology of Legendrian submanifolds in R2n+1, J. Differ. Geom., Volume 71 (2005), pp. 177-305

[12] T. Ekholm; J. Etnyre; M. Sullivan Legendrian contact homology in P×R, Trans. Amer. Math. Soc., Volume 359 (2007), pp. 3301-3335

[13] T. Ekholm; J. Etnyre; L. Ng; M. Sullivan Knot contact homology, Geom. Topol., Volume 17 (2013), pp. 975-1112

[14] T. Ekholm; J. Etnyre; L. Ng; M. Sullivan Filtrations on the knot contact homology of transverse knots, Math. Ann., Volume 355 (2013), pp. 1561-1591

[15] T. Ekholm; L. Ng; V. Shende A complete knot invariant from contact homology | arXiv

[16] Y. Eliashberg Invariants in contact topology, Berlin (Doc. Math.) (1998), pp. 327-338

[17] G. Faonte A-infinity functors and homotopy theory of DG-categories | arXiv

[18] R. Fenn; M. Jordan-Santana; L. Kauffman Biquandles and virtual links, Topol. Appl., Volume 145 (2004), pp. 157-175

[19] S. Gelfand; R. MacPherson; K. Vilonen Perverse sheaves and quivers, Duke Math. J., Volume 83 (1996), pp. 621-643

[20] P. Goerss; J. Jardine Simplicial Homotopy Theory, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2009

[21] D. Goldschmidt Classical link invariants and the Burau representation, Pac. J. Math., Volume 144 (1990), pp. 277-292

[22] P.S. Hirschhorn Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, AMS, Providence, RI, 2003

[23] M. Hovey Model Categories, Mathematical Surveys and Monographs, vol. 63, Amer. Math. Soc., Providence, RI, 1999

[24] S. Humphries An approach to automorphisms of free groups and braids via transvections, Math. Z., Volume 209 (1992), pp. 131-152

[25] S. Humphries Action of braid groups on determinantal ideals, compact spaces and a stratification of Teichmüller space, Invent. Math., Volume 144 (2001), pp. 451-505

[26] A. Joyal; R. Street Braided tensor categories, Adv. Math., Volume 102 (1993), pp. 20-78

[27] A. Joyal; R. Street; D. Verity Traced monoidal categories, Math. Proc. Camb. Philos. Soc., Volume 119 (1996), pp. 447-468

[28] M. Kapranov; V. Schechtman Perverse schobers | arXiv

[29] M. Kapranov; V. Schechtman Perverse sheaves and graphs on surfaces | arXiv

[30] M. Kashiwara; P. Schapira Sheaves on Manifolds, Springer-Verlag, Berlin, 1994

[31] B. Keller, Eur. Math. Soc., Zürich (2006), pp. 151-190

[32] W. Magnus Rings of Fricke characters and automorphism groups of free groups, Math. Z., Volume 170 (1980), pp. 91-103

[33] D. Nadler Microlocal branes are constructible sheaves, Sel. Math. New Ser., Volume 15 (2009), pp. 563-619

[34] D. Nadler; E. Zaslow Constructible sheaves and the Fukaya category, J. Amer. Math. Soc., Volume 22 (2009), pp. 233-286

[35] L. Ng Knot and braid invariants from contact homology I, Geom. Topol., Volume 9 (2005), pp. 247-297

[36] L. Ng Knot and braid invariants from contact homology II, Geom. Topol., Volume 9 (2005), pp. 1603-1637

[37] L. Ng Framed knot contact homology, Duke Math. J., Volume 141 (2008), pp. 365-406

[38] L. Ng Combinatorial knot contact homology and transverse knots, Adv. Math., Volume 227 (2011), pp. 2189-2219

[39] L. Ng (Bolyai Soc. Math. Stud.), Volume vol. 26, János Bolyai Math. Soc., Budapest (2014), pp. 485-530

[40] L. Ng; D. Rutherford; V. Shende; S. Sivek; E. Zaslow Augmentations are sheaves | arXiv

[41] N. Reshetikhin; V. Turaev Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., Volume 127 (1990), pp. 1-26

[42] R. Rouquier Categorification of sl2 and braid groups, Contemp. Math., Volume 406 (2006), pp. 137-167

[43] P. Seidel; R. Thomas Braid group actions on derived categories of coherent sheaves, Duke Math. J., Volume 108 (2001), pp. 37-108

[44] P. Selinger A survey of graphical languages for monoidal categories, Lecture Notes in Physics, vol. 813, Springer, 2011, pp. 289-355

[45] V. Shende The conormal torus is a complete knot invariant | arXiv

[46] V. Shende; A. Takeda Symplectic structures from topological Fukaya categories | arXiv

[47] V. Shende; D. Treumann; E. Zaslow Legendrian knots and constructible sheaves | arXiv

[48] G. Tabuada Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris, Volume 340 (2005), pp. 15-19

[49] D. Tamarkin What do dg-categories form?, Compos. Math., Volume 143 (2007), pp. 1335-1358

[50] B. Toën The homotopy theory of dg-categories and derived Morita theory, Invent. Math., Volume 167 (2007), pp. 615-667

[51] V. Turaev The Yang–Baxter equation and invariants of links, Invent. Math., Volume 92 (1988), pp. 527-553

[52] M. Wada Group invariants of links, Topology, Volume 31 (1992), pp. 399-406

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