UC Davis

In this work we investigate the conjectural ADE classification of exact Lagrangian fillings of Legendrian links. We begin by showing that two methods of constructing exact Lagrangian fillings -- Legendrian weaves and decomposable exact Lagrangian cobordisms without Reidemeister I or II moves -- yield Hamiltonian isotopic exact Lagrangian fillings. Using the method of Legendrian weaves, we construct and distinguish exact Lagrangian fillings in D_n-type. We then investigate Legendrian loops, Legendrian isotopies fixing a Legendrian link pointwise at time one. Legendrian loops act on the set of exact Lagrangian fillings by concatenating the trace of the Legendrian isotopy. We investigate this action first in type A_n, and then more generally, leveraging techniques from the theory of cluster algebras and connections to the theory of mapping class groups. In particular, we give a complete description of the orbital structure of the cluster modular group action on exact Lagrangian fillings of Legendrian (2, n) torus links.

Beyond type A_n, we compile and extend known results interpreting Legendrian loops as generators of cluster modular groups for affine and extended affine type cluster algebras. We show that Legendrian loops virtually generate these cluster modular groups. By extending an analogy between cluster modular groups to mapping class groups, we provide new tests for detecting when a Legendrian loop produces infinitely many distinct exact Lagrangian fillings. Finally, we discuss possible avenues towards producing generating sets for cluster modular groups using Legendrian loops.