Inventiones mathematicae

, Volume 207, Issue 3, pp 1031–1133 | Cite as

Legendrian knots and constructible sheaves

  • Vivek Shende
  • David Treumann
  • Eric ZaslowEmail author


We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou–Kashiwara–Schapira to show that the resulting category is invariant under Legendrian isotopies. A subsequent article establishes its equivalence to a category of representations of the Chekanov–Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov–Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in ‘a’ of the HOMFLY polynomial of the braid closure.



We would like to thank Philip Boalch, Frédéric Bourgeois, Daniel Erman, Paolo Ghiggini, Tamás Hausel, Jacob Rasmussen, Dan Rutherford, and Steven Sivek for helpful conversations. The work of DT is supported by NSF-DMS-1206520 and a Sloan Foundation Fellowship. The work of EZ is supported by NSF-DMS-1104779 and by a Simons Foundation Fellowship.


  1. 1.
    Aganagic, M., Ekholm, T., Ng, L., Vafa, C.: Topological strings, D-model, and knot contact homology. Adv. Theor. Math. Phys. 18, 827–956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aiston, A.K., Morton, H.R.: Idempotents of Hecke algebras of Type A. J. Knot Theory Ramif. 7, 463–487 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Behrend, K.: Derived l-adic categories for algebraic stacks, Mem. AMS, vol. 774 (2003)Google Scholar
  4. 4.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1982)MathSciNetGoogle Scholar
  5. 5.
    Bernstein, I., Gelfand, I., Ponomarev, I.: Coxeter functors and Gabriel’s theorem. Uspehi Mat. Nauk 28, 19–33 (1973)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors, Lecture Notes in Mathematics, vol. 1578. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Biquard, O., Boalch, P.: Wild nonabelian Hodge theory on curves. Compos. Math. 140, 179–204 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bourgeois, F., Chantraine, V.: Bilinearised legendrian contact homology and the augmentation category. arXiv:1210.7367
  9. 9.
    Broué, M., Michel, J.: Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées. In: Finite Reductive Groups: Related Structures and Representations, pp. 73–139. Birkhäuser, Boston (1996)Google Scholar
  10. 10.
    de Cataldo, M., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties. Annals Math. 175, 1329–1407 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chekanov, Yu.: Differential algebra of Legendrian links. Invent. Math. 150, 441–483 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chekanov, Y., Pushkar, P.: Combinatorics of fronts of Legendrian links and the Arnol’d 4-conjectures. Russ. Math Surv. 60(1), 95–149 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chongchitmate, W., Ng, L.: An Atlas of Legendrian Knots. arXiv:1010.3997,
  14. 14.
    Corlette, K.: Flat G-bundles with canonical metrics. J. Diff. Geom. 28(3), 361–382 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Deligne, P.: avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Cohomologie étale; SGA 4\(\frac{1}{2}\), Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977)Google Scholar
  16. 16.
    Deligne, P.: La conjecture de Weil, II. Publ. Math. IHES 52, 137–252 (1980)Google Scholar
  17. 17.
    Deligne, P.: Action du groupe des tresses sur une catégorie. Inv. Math. 128, 159–175 (1997)CrossRefGoogle Scholar
  18. 18.
    Diaconescu, D.-E., Hua, Z., Soibelman, Y.: HOMFLY polynomials, stable pairs and motivic Donaldson–Thomas invariants. Commun. Num. Theor. Phys. 6, 517–600 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Diaconescu, D.-E., Shende, V., Vafa, C.: Large N duality, lagrangian cycles, and algebraic knots. Comm. Math. Phys. 319(3), 813–863 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. (3) 55(1), 127–131 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Drinfeld, V.: DG quotients of DG categories. J. Alg. 272, 643–691 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eliashberg, Y.: Invariants in contact topology. Doc. Math. J. DMV Extra Vol. ICM, 327–338 (1998) (electronic)Google Scholar
  23. 23.
    Ekholm, T.: Rational SFT, Linearized legendrian contact homology, and lagrangian floer cohomology. In: “Perspectives in Analysis, Geometry, and Topology. On the Occasion of the 60th Birthday of Oleg Viro”, pp. 109–145. Springer, Berlin (2012)Google Scholar
  24. 24.
    Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in \({{\mathbb{R}}}^{2n+1}\). J. Diff. Geom. 71, 1–174 (2005)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ekholm, T., Etnyre, J., Sullivan, M.: The contact homology of Legendrian submanifolds in \({{\mathbb{R}}}^{2n+1}\). J. Diff. Geom. 71, 177–305 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory, GAFA, : (Tel Aviv, 1999), Geom. Funct. Anal. 2000. Special Volume, Part II, 560–673 (2000)Google Scholar
  27. 27.
    Etnyre, J.: Legendrian and transversal knots. In: Handook of Knot Theory, pp. 105–185. Elsevier, Amsterdam (2005)Google Scholar
  28. 28.
    Fang, B., Liu, C., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. Invent. Math. 186(1), 79–114 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fuchs, D., Rutherford, D.: Generating families and Legendrian contact homology in the standard contact space. J. Topol. 4, 190–226 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Geiges, H.: An Introduction to Contact Topology. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  31. 31.
    Gaiotto, D., Moore, G., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Goresky, M., Macpherson, R.: Stratified Morse Theory. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  33. 33.
    Guillermou, S., Kashiwara, M., Schapira, P.: Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems. Duke Math. J. 161, 201–245 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gukov, S., Schwarz, A., Vafa, C.: Khovanov–Rozansky homology and topological strings. Lett. Math. Phys. 74, 53–74 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Fredrickson, L.: work in progressGoogle Scholar
  36. 36.
    Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174(3), 555–624 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Henry, M.B., Rutherford, D.: Ruling polynomials and augmentations over finite fields. J. Topol. 8, 1–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55, 59–126 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hitchin, N.: The moduli space of special Lagrangian submanifolds. Ann. Sc. Sup. Norm. Pisa Sci. Fis. Mat. 25, 503–515 (1997)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Jin, X.: Holomorphic branes are perverse sheaves. Geom. Top. 19, 1685–1735 (2015)CrossRefzbMATHGoogle Scholar
  41. 41.
    Kálmán, T.: Rulings of Legendrian knots as spanning surfaces. Pac. J. Math. 237, 287–297 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kashiwara, M.: The Riemann–Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 26, 319–365 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1994)Google Scholar
  44. 44.
    Keller, B.: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Keller, B.: On differential graded categories. Int. Congr. Math. 2, 151–190 (2006)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Khovanov, M.: Triply graded link homology and Hochschild homology of Soergel bimodules. Int. J. Math. 18(8) (2007)Google Scholar
  47. 47.
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology II. Geom. Topol. 12, 1387–1425 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Laszlo, Y., Olsson, M.: The six operations for sheaves on Artin stacks II. Publ. Math. IHES 107(1), 169–210 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Laszlo, Y., Olsson, M.: The perverse t-structure on Artin stacks. Math. Zeit. 261(4), 737–748 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lusztig, G.: Character sheaves I. Adv. Math. 56, 193–237 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Maulik, D.: Stable pairs and the HOMFLY polynomial. arXiv:1210.6323
  52. 52.
    May, J.P.: The additivity of traces in triangulated categories. Adv. Math. 163, 34–73 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Maulik, D., Yun, Z.: Macdonald formula for curves with planar singularities. J. für die reine und angew. Math. (Crelle’s J.) 694, 27–48 (2014)MathSciNetzbMATHGoogle Scholar
  54. 54.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  55. 55.
    Migliorini, L., Shende, V.: A support theorem for Hilbert schemes of planar curves. J. Eur. Math. Soc. 15(6), 2353–2367 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. (N.S.) 15, 563–619 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Nadler, D.: Fukaya categories as categorical morse homology. SIGMA 10, 018 (2014)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Nadler, D., Zaslow, E.: Constructible sheaves and the fukaya category. J. Amer. Math. Soc. 22, 233–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Ng, L.: Computable Legendrian invariants. Topology 42(1), 55–82 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Ng, L.: Framed knot contact homology. Duke Math. J. 141, 365–406 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Ng, L., Rutherford, D.: Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra. Algebr. Geom. Topol. 13, 3047–3097 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Ng, L., Rutherford, D., Shende, V., Sivek, S., Zaslow, E.: Augmentations are sheaves. arXiv:1502.04939
  63. 63.
    Ng, L., Rutherford, D., Shende, V., Sivek, S.: The cardinality of the augmentation category of a Legendrian link. arXiv:1511.06724
  64. 64.
    Ng, L., Sabloff, J.: The correspondence between augmentations and rulings for Legendrian knots. Pac. J. Math. 224, 141–150 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Oblomkov, A., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link. Duke Math. J. 161(7), 1277–1303 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Oblomkov, A., Rasmussen, J., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. arXiv:1201.2115
  67. 67.
    Oblomkov, A., Yun, Z.: Geometric representations of graded and rational Cherednik algebras. arXiv:1407.5685
  68. 68.
    Rennemo, J.: Homology of Hilbert schemes of points on a locally planar curve. arXiv:1308.4104
  69. 69.
    Rouquier, R.: Categorification of the braid group. arXiv:math.RT/0409593
  70. 70.
    Rutherford, D.: The Thurston–Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond. Int. Math. Res. Notices 2006, 1–15 (2006). doi: 10.1155/IMRN/2006/78591
  71. 71.
    Saito, M.: Mixed Hodge modules and applications. In Proceedings of the International Congress of Mathematicians, (Kyoto, 1990) vol. I, II, (Math. Soc. Japan, Tokyo, 1991), pp. 725–734Google Scholar
  72. 72.
    Schmid, W., Vilonen, K.: Characteristic cycles of constructible sheaves. Invent. Math. 124, 451–502 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Schürmann, J.: Topology of singular spaces and constructible sheaves. Monografie Matematyczne, vol. 63. Birkhäuser, Boston (2003)Google Scholar
  74. 74.
    Shende, V.: Generating families and constructible sheaves. arXiv:1504.01336
  75. 75.
    Shende, V.: The wild character variety of an irregular singularity and the HOMFLY homology of its asymptotic link (in preparation)Google Scholar
  76. 76.
    Shende, V., Treumann, D., Williams, H., Zaslow, E.: Cluster varieties from Legendrian knots. arXiv:1512.08942
  77. 77.
    Shepard, A.: A cellular description of the derived category of a stratified space. Phd. Thesis, Brown University (1985)Google Scholar
  78. 78.
    Simpson, C.T.: Higgs bundles and local systems. Publ. Math. IHES 75, 5–95 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Simpson, C.T.: Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3, 713–770 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Sivek, S.: A bordered Chekanov–Eliashberg algebra. J. Topol. 4(1), 73–104 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Tamarkin, D.: Microlocal condition for non-displaceability. arXiv:0809.1584
  82. 82.
    Traynor, L.: Generating function polynomials for Legendrian links. Geom. Top. 5, 719–760 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Toën, B.: Higher and derived stacks, a global overview. arXiv:math/0604504
  84. 84.
    Webster, B., Williamson, G.: A geometric model for Hoshschild homology of Soergel bimodules. Geom. Topol. 12, 1243–1263 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Webster, B., Williamson, G.: A geometric construction of colored HOMFLYPT homology. arXiv:0905.0486
  86. 86.
    Webster, B., Williamson, G.: The geometry of Markov traces. arXiv:0911.4494
  87. 87.
    Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Witten, E.: Khovanov homology and Gauge theory. arXiv:1108.3103
  89. 89.
    Witten, E.: Gauge theory and wild ramification. arXiv:0710.0631

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Authors and Affiliations

  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsNorthwestern UniversityEvanstonUSA

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