Constructing Lagrangians from triple grid diagrams

TL;DR

This paper introduces triple grid diagrams, a new combinatorial tool that encodes Legendrian links and constructs Lagrangian surfaces in complex projective space, with potential applications in symplectic topology.

Contribution

It defines and studies triple grid diagrams, linking them to Lagrangian surfaces in , and shows how they can be used to construct Lagrangian caps and surfaces.

Findings

Triple grid diagrams encode Legendrian links in contact 3-spheres.

They determine Lagrangian caps in  with Legendrian boundary.

Examples of triple grid diagrams are constructed and analyzed.

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.