Moduli Spaces of Lagrangian Surfaces in $\mathbb{CP}^2$ obtained from Triple Grid Diagrams

TL;DR

This paper introduces a geometric construction for the moduli space of triple grid diagrams, which encode Lagrangian surfaces in complex projective space, simplifying their classification through linear algebra techniques.

Contribution

It provides a novel geometric method to explicitly construct and analyze the moduli space of triple grid diagrams, improving upon previous approximation and search methods.

Findings

Moduli space can be described via a geometric construction.

Problem reduces to linear algebra, enabling polynomial-time solutions.

Facilitates explicit examples of Lagrangian surfaces in $\,\mathbb{CP}^2$.

Abstract

Links in $S^3$ as well as Legendrian links in the standard tight contact structure on $S^3$ can be encoded by grid diagrams. These consist of a collection of points on a toroidal grid, connected by vertical and horizontal edges. Blackwell, Gay and second author studied triple grid diagrams, a generalization where the points are connected by vertical, horizontal and diagonal edges. In certain cases, these determine Lagrangian surfaces in $\mathbb{CP}^2$. However, it was difficult to construct explicit examples of triple grid diagrams, either by an approximation method or combinatorial search. We give an elegant geometric construction that produces the moduli space of all triple grid diagrams. By conditioning on the abstract graph underlying the triple grid diagram, as opposed to the grid size, the problem reduces to linear algebra and can be solved quickly in polynomial time.