Constructing Lagrangians from triple grid diagrams

TL;DR

The paper develops a combinatorial framework—triple grid diagrams on a torus—to encode Lagrangian surfaces in CP^2 via three Legendrian links in copies of S^3. It proves that a triple grid diagram yields a Lagrangian cap L(D) in CP^2 minus three balls, with boundary the associated Legendrian links, and that when these links are tb=-1 unlinks, L(D) can be capped to produce a closed Lagrangian in CP^2. The authors build the cap through a detailed construction: converting grid-point data into Legendrian arcs in solid tori, gluing to form Legendrian links in S^3, and assembling a CP^2 cap using flows along Liouville and symplectic vector fields, organized by a moment-map decomposition. They provide extensive examples and discuss obstructions to fillability, connecting topological invariants of the triple grid diagrams to the existence of embedded or immersed Lagrangian surfaces in CP^2, and outlining directions for future work on uniqueness and moves between diagrams. The work lays groundwork for systematically obstructing or realizing Lagrangian fillings via discrete combinatorial data.

Abstract

Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of Legendrian links in the standard contact 3-sphere. In this paper, we define and investigate triple grid diagrams, a generalization to toroidal diagrams consisting of horizontal, vertical, and diagonal grid lines. In certain cases, a triple grid diagram determines a closed Lagrangian surface in $\mathbb{CP}^2$. Specifically, each triple grid diagram determines three grid diagrams (row-column, column-diagonal and diagonal-row) and thus three Legendrian links, which we think of collectively as a Legendrian link in a disjoint union of three standard contact 3-spheres. We show that a triple grid diagram naturally determines a Lagrangian cap in the complement of three Darboux balls in $\mathbb{CP}^2$, whose negative boundary is precisely this Legendrian link. When these Legendrians are maximal Legendrian unlinks, the Lagrangian cap can be filled by Lagrangian slice disks to obtain a closed Lagrangian surface in $\mathbb{CP}^2$. We construct families of examples of triple grid diagrams and discuss potential applications to obstructing Lagrangian fillings.

Figures (23)

• Figure 1: A genus one Heegaard splitting of $S^3$ (left) and a genus one trisection of $\mathbb{CP}^{2}$ (right).
• Figure 2: Some examples of combinatorial triple grid diagrams.
• Figure 3: A (combinatorial) triple grid diagram. The right-hand side shows separately the three grids making up the triple grid diagram on the left-hand side, along with the knot represented by each grid. Colors are used to indicate where each vertex in the triple grid diagram shows up in the three individual grids. Below the grids the process for obtaining Legendrians is shown.
• Figure 4: The trivalent graph $\Gamma(D)$ associated to the triple grid diagram $D$ shown in \ref{['fig:grid_Leg']}. The left-hand side depicts the graph still on the grid, with the green edges wrapping around the torus, while the right-hand side depicts the graph abstractly.
• Figure 5: Arcs on meridional disks in a solid torus $H$, with radial coordinate $p$, meridional coordinate $\mu$ and longitudinal coordinate $\lambda$. The torus is illustrated as a cylinder, with the understanding that the top is glued to the bottom by the identity.

Theorems & Definitions (46)

• Definition 1.1
• Definition 1.2
• Theorem 1.1: Executive summary
• Corollary 1.1: Executive summary
• Theorem 1.3: Restated more precisely
• Corollary 1.4: Restated more precisely
• proof
• Remark 1.1
• Corollary 1.6
• proof