Classification of Width 1 Lattice Tetrahedra by Their Multi-Width

Abstract

We introduce the multi-width of a lattice polytope and use this to classify and count all lattice tetrahedra with multi-width \((1,w_2,w_3)\). The approach used in this classification can be extended into a computer algorithm to classify lattice tetrahedra of any given multi-width. We use this to classify tetrahedra with multi-width \((2,w_2,w_3)\) for small \(w_2\) and \(w_3\) and make conjectures about the function counting lattice tetrahedra of any multi-width.