Abstract: In this talk, we compare two different approaches to constructing Lagrangian fillings of Legendrian knots. The first one is conjugate Lagrangian fillings of alternating Legendrians, introduced by Shende-Treumann-Williams-Zaslow, which are characterized using bipartite graphs, and the second one is Lagrangian projections of Legendrian weaves, introduced by Casals-Zaslow, which are depicted by planar graphs encoding their wavefronts. We will develop a diagrammatic calculus to show that conjugate Lagrangian fillings are Hamiltonian isotopic to certain Lagrangian projections of Legendrian weaves. The result includes Legendrian positive braid closures and ideal triangulations on punctured surfaces. We will then explain some implications on Lagrangian mutations and cluster theory. This is joint work in preparation with Roger Casals.
