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Inventiones mathematicae

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

Legendrian knots and constructible sheaves

  • Vivek Shende
  • David Treumann
  • Eric ZaslowEmail author
Article

Abstract

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.

Notes

Acknowledgments

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.

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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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