Cubic Planar Graphs and Legendrian Surface Theory

TL;DR

This paper explores Legendrian surfaces associated with cubic planar graphs, revealing how their properties relate to graph invariants, sheaf moduli, and non-exact fillings, with implications for symplectic and Gromov-Witten theories.

Contribution

It introduces a novel connection between cubic planar graphs and Legendrian surfaces, demonstrating how graph invariants influence Legendrian isotopy classes and their fillings.

Findings

Graphs with different chromatic polynomials produce non-isotopic Legendrian surfaces.

Legendrian surfaces studied have no exact Lagrangian fillings but possess interesting non-exact fillings.

The sheaf moduli space is described via the graph and embeds as a holomorphic Lagrangian in a symplectic domain.

Abstract

We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural open Gromov-Witten invariants for the non-exact filling, following Aganagic-Vafa [AV, AV2].