Cluster varieties from Legendrian knots

Abstract

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

arXiv comment47 pages

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Cited by in corpus (40)

• Legendrian knots and constructible sheaves
• Microlocal Morse theory of wrapped Fukaya categories
• Non-squeezing property of contact balls
• On the combinatorics of exact Lagrangian surfaces
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• The conormal torus is a complete knot invariant
• Lagrangian pairs of pants
• Morsifications and mutations
• Schubert puzzles and integrability I: invariant trilinear forms
• Calabi-Yau structures on topological Fukaya categories
• Brane structures in microlocal sheaf theory
• Augmentations, Fillings, and Clusters
• Sheaf quantization in Weinstein symplectic manifolds
• Infinitely many Lagrangian fillings
• Domain wall formation and magnon localization in twisted chromium trihalides
• Algebras of quantum monodromy data and decorated character varieties
• Lagrangian submanifolds from tropical hypersurfaces
• Weave Realizability for D-type
• Critical varieties in the Grassmannian
• Tropical Lagrangians in toric del-Pezzo surfaces
• Augmentations and immersed Lagrangian fillings
• Lagrangian fillings for Legendrian links of affine type
• Fukaya category of Grassmannians: rectangles
• Exotic Lagrangian tori in Grassmannians
• Positive Braid Links with Infinitely Many Fillings
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• Wrapped sheaves
• Integrability As Duality: The Gauge/YBE Correspondence
• Augmentations and ruling polynomials for Legendrian graphs
• A Lagrangian filling for every cluster seed
• On the equatorial Dehn twist of a Lagrangian nodal sphere
• A Legendrian Turaev torsion via generating families
• Twisted spectral correspondence and torus knots
• Smoothly non-isotopic Lagrangian disk fillings of Legendrian knots
• Categorification of Legendrian knots
• Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop
• Non-fillable augmentations of twist knots
• Lagrangian cobordism of positroid links
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Symplectomorphisms of some Weinstein 4-manifolds