Constructing Lagrangian surfaces in all symplectic toric $4$-manifolds from multi-grid diagrams

Abstract

We present a new diagrammatic construction of Lagrangian surfaces in all symplectic toric $4$-manifolds from objects called multi-grid diagrams. These are a generalisation of triple grid diagrams for Lagrangian surfaces in $\mathbb{CP}^2$ wherein the underlying trisection is replaced by the more general notion of a toric decomposition. We give examples of multi-grid diagrams for Lagrangian surfaces in a range of symplectic toric $4$-manifolds, including Evans' recent non-squeezable Klein bottle in $S^2 \times S^2$.