Rational polygons with no interior lattice points

This record contains maximal \(k\)-rational polygons with no interior lattice points for \(1 \leq k \leq 21\), all \(k\)-rational polygons with no interior lattice points, except those that can be realized in the strip \( \mathbb{R} \times [0,1] \) for \(1 \leq k \leq 6\), all Ehrhart quasipolynomials of these polygons for \(1 \leq k \leq 6\). The maximal polygons are stored as text files with one polygon per line. Each line contains the vertices of the polygon multiplied by \(k\) (so that they are integral). For \(1 \leq k \leq 6\), the files that contain all polygons as well as the Ehrhart quasipolynomials are stored using the HDF5 file format. All polygons and Ehrhart quasipolynomials have been obtained using RationalPolygons.jl. Structure of the HDF5 files for polygons In the HDF5 files "all_k<k>.h5", the polygons are split into different datasets according to their normalized area and number of vertices. We use "a" to denote the normalized area (twice the euclidian area) and "n" to denote the number of vertices. For example, the \(3\)-rational polygons with normalized area \(30\) having \(5\) vertices are located in "all_k3.h5" under the dataset "/a30/n5". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers. These integers are the vertices of the polygon multiplied by \(k\), stored in column major layout. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((k\cdot x_1,k\cdot y_1,k \cdot x_2,k\cdot y_2,k \cdot x_3,k \cdot y_3)\). Moreover, each file contains the special two-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for a given normalized volume and number of vertices. Structure of the HDF5 files for Ehrhart quasipolynomials In the HDF5 files "ehrhart_k<k>.h5", the Ehrhart quasipolynomials are split into different datasets according to the normalized area of the associated polygon. We use "a" for the normalized area (twice the euclidian area). For example, the Ehrhart quasipolynomials of all \(3\)-rational polygons with normalized area \(30\) are located in "ehrhart_k3.h5" under the dataset "/a30". Each Ehrhart quasipolynomial is stored as a \(3 \times k\) integral matrix. Hence each dataset is three-dimensional, with the first dimension to enumerate the quasipolynomials and the latter two dimensions to store each quasipolynomial itself. For the way Ehrhart quasipolynomials are encoded as \(3 \times k\) integral matrices, we refer to the relevant documentation of RationalPolygons.jl.