Polygons with small denominator containing a small number of lattice points

The denominator of a rational polytope \(P\) is an integer \(r\) such that the dilated polytope \(rP\) has lattice point vertices. The size of a polytope is the number of lattice points it contains. This dataset contains polygons with denominator 2 and 3 with small size, classified using a growing algorithm as described in [HHK24].

The data consists of files "denom_r_size_k_polygons.txt" which record the denominator \(r\) size \(k\) polygons \(P\). Each entry consists of the vertices and volume of \(rP\), the Ehrhart \(\delta\)-vector/\(h^*\)-vector of \(P\), and an ID number, which is unique among polygons of given size and denominator. Entries are ordered by their ID number. There are 50,564 entries in total.

Example entry:

ID=1
Vertices=[[ 1, 0 ], [ 0, 1 ], [ 3, 5 ]]
Volume=7
DeltaVec=[ 1, 0, 3, 7, 3, 0 ]

If you make use of this data, please cite [HHK24] and the DOI for this data:

doi:10.5281/zenodo.14230584

References:

[HHK24] Girtrude Hamm, Johannes Hofscheier, Alexander Kasprzyk, Classification and Ehrhart Theory of Denominator 2 Polygons. (preprint) arxiv:2411.19183