Legendrian knots and constructible sheaves

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, and conjecture it is equivalent to the representation category 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.

92 pages, final journal version, Inventiones Mathematicae (2016)

References in corpus (9)

• Higher and derived stacks: a global overview
• Cluster varieties from Legendrian knots
• Categorification of the braid groups
• Microlocal condition for non-displaceablility
• Gauge Theory And Wild Ramification
• A bordered Chekanov-Eliashberg algebra
• Holomorphic Lagrangian branes correspond to perverse sheaves
• The cardinality of the augmentation category of a Legendrian link
• Generating families and constructible sheaves

Cited by in corpus (44)

• Cluster varieties from Legendrian knots
• Twisted wild character varieties
• Non-squeezing property of contact balls
• On the combinatorics of exact Lagrangian surfaces
• The conormal torus is a complete knot invariant
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• Augmentations are Sheaves
• Calabi-Yau structures on topological Fukaya categories
• Generating families and augmentations for Legendrian surfaces
• Brane structures in microlocal sheaf theory
• Mirror symmetry for the Landau-Ginzburg A-model $M={\mathbb{C}}^n,W=z_1\cdots z_n$
• Sheaf quantization in Weinstein symplectic manifolds
• Domain wall formation and magnon localization in twisted chromium trihalides
• Infinitely many Lagrangian fillings
• The three cusps conjecture
• Weave Realizability for D-type
• Cluster Structures on Double Bott-Samelson Cells
• The cardinality of the augmentation category of a Legendrian link
• Lagrangian fillings for Legendrian links of affine type
• Legendrian skein algebras and Hall algebras
• Sheaves via augmentations of Legendrian surfaces
• BPS states, torus links and wild character varieties
• Augmentations are sheaves for Legendrian graphs
• Generating families and constructible sheaves
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• Wrapped sheaves
• Categorical localization for the coherent-constructible correspondence
• Ruling polynomials and augmentations for Legendrian tangles
• A Lagrangian filling for every cluster seed
• Representations, sheaves, and Legendrian $(2,m)$ torus links
• Radon Transform for Sheaves
• Twisted spectral correspondence and torus knots
• Eisenstein series via factorization homology of Hecke categories
• A Legendrian Turaev torsion via generating families
• Perverse Sheaves and Knot Contact Homology
• Augmentations and link group representations
• Algebra and geometry of link homology
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Augmented Legendrian cobordism in $J^1{S}^1$
• Lagrangian cobordism of positroid links
• Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on ${\mathbb{C}}^2$
• Augmentations and sheaves for links
• Non-fillable augmentations of twist knots
• Arboreal singularities and loose Legendrians I