A Census of Convex Lattice Polygons with at most one

We find all convex lattice polygons in the plane (up to equivalence) with at most one interior lattice point. A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. In 1980, Arkinstall [1] proved that, up to lattice equivalence (defined below), there is just one convex lattice hexagon containing a single interior lattice point. In this article, we will extend this result by finding all convex lattice polygons with at most one interior lattice point. Having this characterization will help in proving inequalities about lattice polygons containing just one interior lattice point which in turn may help in the investigation of general convex bodies in the plane with lattice point constraints (see [3], [4], and [5]). To understand when two lattice polygons are “equivalent”, we must first review some definitions concerning standard transformations of the plane. An affine transformation is a linear transformation followed by a translation. A unimodular transformation is one that preserves area. To be unimodular, the matrix corresponding to a linear transformation must have determinant 1. If furthermore, the entries of the matrix are integers, then the transformation is known as an integral unimodular transformation and has the property that it preserves the number of lattice points in a set. An integral unimodular affine transformation (also known as an equiaffinity) in the plane can be expressed by the 3 × 3 matrix in the equation   a b e c d f 0 0 1     x y 1   =   x′ y′ 1   where a, b, c, d, e, and f are integers and ad−bc = 1. This includes an integral translation by the vector < e, f >. Two lattice polygons are said to be lattice equivalent if one can be transformed into the other via an integral unimodular affine transformation. A shear is an integral unimodular transformation that leaves all the points on a line fixed. It is also referred to as a shear about this line. In the plane, a shear about the x-axis Reprinted from Ars Combinatoria 28(1989)83–96