Lagrangian Skeleta and Plane Curve Singularities

Abstract

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed arboreal Lagrangian skeleta for Weinstein pairs and Weinstein 4-manifolds associated to max-tb Legendrian representatives of algebraic links. We provide computations of Legendrian and Weinstein invariants, and discuss the symplectic topological nature of the Fomin-Pylyavskyy-Shustin-Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.

27 Pages, 11 Figures