  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.

Geometric Topology;Symplectic Geometry