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Lagrangian Fillings in A-type and their Kalman Loop Orbits

James Hughes

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.

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 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.

Paper Structure

This paper contains 31 sections, 21 theorems, 34 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Equivalence of Lagrangian fillings
  3. Kálmán loop orbits
  4. Preliminaries on Lagrangian fillings and their invariants
  5. Legendrian weaves
  6. $N$-Graphs and Singularities of Fronts
  7. The $D_4^-$ cobordism
  8. Vertical weaves
  9. 2-graphs and weave fillings of $\lambda(A_{n-1})$
  10. The Legendrian contact DGA and the augmentation variety
  11. The pinching cobordism and pinching sequence fillings
  12. The Legendrian contact DGA
  13. Augmentations
  14. Braid matrices
  15. Continuants
  16. ...and 16 more sections

Key Result

Theorem \oldthetheorem

For any exact Lagrangian filling of $\lambda(\beta)$ constructed via a sequence of pinching cobordisms and traces of Reidemeister III moves, there is unique a Hamiltonian isotopic weave filling.

Figures (17)

  • Figure 1: Front projections of the Legendrian isotopic links given as the rainbow closure (left) and $(-1)$-framed closure (right) of the positive braids $\beta$ and $\beta\Delta^2$. Here $\Delta$ denotes a half twist of the braid.
  • Figure 2: Singularities of front projections of Legendrian surfaces. Labels correspond to notation used by Arnold in his classification.
  • Figure 3: The weaving of the singularities pictured in Figure \ref{['fig: wavefronts']} along the edges of the $N$-graph. Gluing these local pictures together according to the $N$-graph $\Gamma$ yields the weave $\Lambda(\Gamma)$.
  • Figure 4: A local model of a generic perturbation of the $D_4^-$ singularity as the front projection of a Legendrian surface (top) and as a movie of 1-dimensional fronts (bottom). The Reeb chord is depicted as a dashed red line. We first apply a Reidemeister I move before adding a 1-handle and applying two more Reidemeister I moves to arrive at a diagram with a single crossing.
  • Figure 5: A pair of 2-graphs representing the same weave filling of $\lambda(A_5)$. On the left, the 2-graph $\Gamma$ is inscribed in its dual triangulation of the octagon. On the right, the corresponding vertical weave is the image of the diffeomorphism $\varphi$. The edges of the vertical 2-graph are labeled by the nearest counterclockwise label of the dual triangulation. The dotted lines on the left give a foliation of $D^2$, corresponding to the foliation of $\mathbb{R}\times (-\infty, 0]$ depicted on the right.
  • ...and 12 more figures

Theorems & Definitions (58)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example
  • Proposition 2.1
  • ...and 48 more