Affine dimers from characteristic polygons

TL;DR

This paper classifies convex lattice polygons that can serve as characteristic polygons of affine dimers, providing constructions, algorithms, and proving existence for certain classes of polygons.

Contribution

It introduces methods to construct affine dimers from existing ones, and proves existence results for polygons of specific types.

Findings

All lattice triangles admit an affine dimer.

Generalized parallelograms admit an affine dimer.

Polygons of genus at most two admit an affine dimer.

Abstract

Recent work by Forsg{\aa}rd indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the classification of convex lattice polygons arising as characteristic polygons of affine dimers. We present several general constructions of new affine dimers from old, and an algorithm for finding affine dimers with prescribed polygon. With these tools we prove that all lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.