Lagrangian Fillings in Atype and their Kalman Loop Orbits
Abstract
We continue the study of exact Lagrangian fillings of Legendrian (2,n) torus links, as first initiated by EkholmHondaKalman and Pan. Our main result proves that for a decomposable exact Lagrangian filling described through a pinching sequence, there exists a unique weave filling in the same Hamiltonian isotopy class. As an application of this result we describe the orbital structure of the Kalman loop and give a combinatorial criteria to determine the orbit size of a filling. We first give a Floertheoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. This is followed by an alternative geometric proof of the orbital structure, obtained as a corollary of the main result. We conclude by giving a purely combinatorial description of the Kalman loop action on the fillings discussed above in terms of edge flips of triangulations.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.09662
 Bibcode:
 2021arXiv210909662H
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 53D12;
 57K33
 EPrint:
 27 pages, 10 figures