Decompositions of augmentation varieties via weaves and rulings

An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. For certain Legendrian knots, the moduli space of fillings also known as an augmentation variety is isomorphic to a braid variety. Both admit decompositions, one coming from weaves and one from normal graded rulings (a Legendrian invariant). In joint work with Asplund, Hughes, Leverson, Li and Wu we prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety.