### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1031-1133 |

Number of pages | 103 |

Journal | Inventiones Mathematicae |

Volume | 207 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*207*(3), 1031-1133. https://doi.org/10.1007/s00222-016-0681-5

}

*Inventiones Mathematicae*, vol. 207, no. 3, pp. 1031-1133. https://doi.org/10.1007/s00222-016-0681-5

**Legendrian knots and constructible sheaves.** / Shende, Vivek; Treumann, David; Zaslow, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Legendrian knots and constructible sheaves

AU - Shende, Vivek

AU - Treumann, David

AU - Zaslow, Eric

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84981205587&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84981205587&partnerID=8YFLogxK

U2 - 10.1007/s00222-016-0681-5

DO - 10.1007/s00222-016-0681-5

M3 - Article

VL - 207

SP - 1031

EP - 1133

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -