Lagrangian fillings in $A$-type and their Kálmán loop orbits

Abstract

We compare two constructions of exact Lagrangian fillings of Legendrian positive braid closures, the Legendrian weaves of Casals–Zaslow, and the decomposable Lagrangian fillings of Ekholm–Honda–Kálmán, and show that they coincide for large families of Lagrangian fillings. As a corollary, we obtain an explicit correspondence between Hamiltonian isotopy classes of decomposable Lagrangian fillings of Legendrian $(2,n)$ torus links described by Ekholm–Honda–Kálmán and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the Kálmán loop and give a combinatorial criteria to determine the orbit size of a filling. We follow our geometric discussion with a Floer-theoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. We conclude by giving a purely combinatorial description of the Kálmán loop action on the fillings discussed above in terms of edge flips of triangulations.