We first show that both the conjugated Lagrangian filling associated with the Pravic graph and the Lagrangian filling obtained as the Reeb pinch sequence are Hamiltonian isotopes for the Lagrangian projection of the legend weave. In general, we establish a new set of Leidemeister movements for hybrid Lagrangian surfaces. These allow for explicit combinatorial isotopes between different types of Lagrangian fillings, and using them, Legend Weave actually generalizes these previously known combinatorial methods to construct Lagrangian fillings. Indicates that This generalization is strict. Textiles can typically generate an infinite number of Hamiltonian isotope classes for Lagrange fillings, whereas conjugate planes and Reeve pinch sequences generate a finite number of fillings.
We then compare the layer quantizations associated with each such type of Lagrangian filling and show that the cluster structures of the corresponding moduli of quasi-perfect objects agree. In particular, this shows that the clustering variables of the Bott-Sammerson cell given as generalized minor are geometric microlocal holonomy associated with layer quantization. Similar results are shown for his Fock-Goncharov cluster variables in the moduli space of framed local systems. In the course of this article and its appendix, we also establish some technical results necessary for rigorous comparisons between different Lagrangian fillings and their microlocal sheaf invariants.
