Lagrangian skeleta and plane curve singularities (Q6601733)

This paper investigates a relation between the theory of isolated plane curve singularities, as developed by Arnol'd, Gusein-Zade, A'Campo, Milnor and others, and arboreal Lagrangian skeleta of Weinstein 4-manifolds.\N\NSuppose that the function \(f:\mathbb{C}^2\rightarrow \mathbb{C}\) defines an isolated plane curve singularity at the origin. The author constructs closed Lagrangian skeleta for the infinite class of Weinstein 4-manifolds obtained by attaching Weinstein 2-handles to the link of \(f\). 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 \(\Lambda \subseteq (\mathbb{S}^3,\xi_{\mathrm{st}})\).\N\NThe author provides computations of Legendrian and Weinstein invariants, and discusses the contact topological nature of the Fomin-Pylyavskyy-Shustin-Thurston cluster algebra associated to a singularity. He presents a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links (the series \(A_n\) and \(D_n\) and the individual singularities \(E_6\), \(E_7\) and \(E_8\) are the simple singularities).\N\NThe author also lists some related problems.\N\NFor the entire collection see [Zbl 1515.53004].