Rounding corners of polygons and the embedded contact homology of $T^3$

Abstract

The embedded contact homology (ECH) of a 3–manifold with a contact form is a variant of Eliashberg–Givental–Hofer’s symplectic field theory, which counts certain embedded $J$–holomorphic curves in the symplectization. We show that the ECH of $T^3$ is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves “rounding corners”. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsváth–Szabó Floer homology $H{F}^{+}\left(T^3\right)$.