Legendrian knots and constructible sheaves

Vivek Shende, David Treumann, Eric Zaslow

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1031-1133
Number of pages103
JournalInventiones Mathematicae
Volume207
Issue number3
DOIs
StatePublished - Mar 1 2017

Fingerprint

Legendrian Knot
Constructible
Sheaves
Braid
Closure
Moduli Space
Knot
HOMFLY Polynomial
Differential Graded Algebra
Knot Theory
Local System
Cotangent Bundle
Spectral Sequence
Ring or annulus
Branes
Filtration
Galois field
Homology
Lowest
Circle

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Shende, Vivek ; Treumann, David ; Zaslow, Eric. / Legendrian knots and constructible sheaves. In: Inventiones Mathematicae. 2017 ; Vol. 207, No. 3. pp. 1031-1133.
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Legendrian knots and constructible sheaves. / Shende, Vivek; Treumann, David; Zaslow, Eric.

In: Inventiones Mathematicae, Vol. 207, No. 3, 01.03.2017, p. 1031-1133.

Research output: Contribution to journalArticle

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