Classification of lattice triangles by their two smallest widths

TL;DR

This paper introduces the second lattice width concept to classify lattice triangles by their widths, explores their automorphisms and Ehrhart theory, and connects lattice triangle enumeration to algebraic geometry via generating functions.

Contribution

It defines the second lattice width, classifies lattice triangles by widths, and links enumeration to Hilbert series of a hypersurface, advancing lattice polytope theory.

Findings

Classification of lattice triangles by width and second width.

Automorphism groups and Ehrhart properties of classified triangles.

Generating function for counting lattice triangles relates to a hypersurface Hilbert series.

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 $\mathbb{P}(1,1,1,2,2,2)$.