Classification of lattice triangles by their two smallest widths

Abstract

We introduce the notion of the second lattice width of a lattice polytope and use this to classify lattice triangles by their width and second width. This is equivalent to classifying lattice triangles contained in a given rectangle (and no smaller rectangle) up to affine equivalence. Using this classification we investigate the automorphism groups and Ehrhart theory of lattice triangles. We also show that the sequence counting lattice triangles contained in dilations of the unit square has generating function equal to the Hilbert series of a degree 8 hypersurface in $\mathbb{P}(1,1,1,2,2,2)$.