Lagrangian Skeleta and Plane Curve Singularities

TL;DR

This paper constructs and analyzes Lagrangian skeleta related to plane curve singularities, connecting symplectic topology, Legendrian invariants, and cluster algebras, with conjectures on Lagrangian fillings.

Contribution

It introduces a method to build arboreal Lagrangian skeleta for singularities and Weinstein manifolds, linking them to Legendrian invariants and cluster algebra structures.

Findings

Construction of arboreal Lagrangian skeleta for plane curve singularities

Computation of Legendrian and Weinstein invariants

Proposed ADE-classification conjecture for Lagrangian fillings

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.