Conjugate Fillings and Legendrian Weaves

First, we show that conjugate Lagrangian fillings, associated to plabic graphs, and Lagrangian fillings obtained as Reeb pinching sequences are both Hamiltonian isotopic to Lagrangian projections of Legendrian weaves. In general, we establish a series of new Reidemeister moves for hybrid Lagrangian surfaces. These allow for explicit combinatorial isotopies between the different types of Lagrangian fillings and we use them to show that Legendrian weaves indeed generalize these previously known combinatorial methods to construct Lagrangian fillings. This generalization is strict, as weaves are typically able to produce infinitely many distinct Hamiltonian isotopy classes of Lagrangian fillings, whereas conjugate surfaces and Reeb pinching sequences produce finitely many fillings. Second, we compare the sheaf quantizations associated to each such types of Lagrangian fillings and show that the cluster structures in the corresponding moduli of pseudo-perfect objects coincide. In particular, this shows that the cluster variables in Bott-Samelson cells, given as generalized minors, are geometric microlocal holonomies associated to sheaf quantizations. Similar results are presented for the Fock-Goncharov cluster variables in the moduli spaces of framed local systems. In the course of the article and its appendices, we also establish several technical results needed for a rigorous comparison between the different Lagrangian fillings and their microlocal sheaf invariants.