Abstract
We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs (mathbb {C}^2,Lambda ) and Weinstein 4-manifolds W(Lambda ) associated to max-tb Legendrian representatives of algebraic links Lambda subseteq (mathbb {S}^3,xi _text {st}). We provide computations of Legendrian and Weinstein invariants, and discuss the contact 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.
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