Lagrangian Fillings in A-type and their Kalman Loop Orbits

TL;DR

This paper establishes a correspondence between two constructions of Lagrangian fillings of Legendrian links, analyzes their orbit structure under the Kálmán loop, and provides combinatorial and Floer-theoretic insights into their symmetries.

Contribution

It demonstrates the equivalence of Legendrian weave and decomposable Lagrangian fillings for large families and describes the Kálmán loop orbits using combinatorics and Floer theory.

Findings

Explicit correspondence between different Lagrangian fillings.

Description of the Kálmán loop orbit structure.

A combinatorial criterion for orbit size.

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-K\'alm\'an 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-K\'alm\'an and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the K\'alm\'an 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 K\'alm\'an loop action on the fillings discussed above in terms of edge flips of triangulations.