  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.
