Augmentations, Fillings, and Clusters

Abstract

We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K2 (aka. A-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

62 pages. The new version combines the previous version and 2009.00499

References in corpus (8)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Cluster algebras IV: Coefficients
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• Augmentations and Rulings of Legendrian Knots
• Cluster algebras III: Upper bounds and double Bruhat cells
• Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology
• Constructible Sheaves and the Fukaya Category
• Braid-positive Legendrian links

Cited by in corpus (8)

• Weave Realizability for D-type
• A note on infinite number of exact Lagrangian fillings for spherical spuns
• Lagrangian fillings for Legendrian links of affine type
• Positive Braid Links with Infinitely Many Fillings
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• Algebra and geometry of link homology
• Non-fillable augmentations of twist knots
• Lagrangian cobordism functor in microlocal sheaf theory I