Lagrangian Fillings in A-type and their Kalman 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.09662
- arXiv:
- arXiv:2109.09662
- Bibcode:
- 2021arXiv210909662H
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics;
- 53D12;
- 57K33
- E-Print:
- 36 pages, 17 figures. v2 updates based on referee feedback