Non-spanning lattice 3-polytopes

TL;DR

This paper classifies non-spanning lattice 3-polytopes, revealing their structure, properties of empty tetrahedra, and computing their h*-vectors, thus advancing understanding of lattice polytopes' combinatorial geometry.

Contribution

It provides a complete classification of non-spanning lattice 3-polytopes and describes their structure, including properties of empty tetrahedra and h*-vectors, with implications for spanning polytopes.

Findings

Most non-spanning 3-polytopes have a simple structure with specific lattice point arrangements.

All empty tetrahedra in these polytopes have the same volume and form a triangulation.

Spanning 3-polytopes generally contain a unimodular tetrahedron, with two exceptions.

Abstract

We completely classify non-spanning $3$-polytopes, by which we mean lattice $3$-polytopes whose lattice points do not affinely span the lattice. We show that, except for six small polytopes (all having between five and eight lattice points), every non-spanning $3$-polytope $P$ has the following simple description: $P\cap \mathbb{Z}^3$ consists of either (1) two lattice segments lying in parallel and consecutive lattice planes or (2) a lattice segment together with three or four extra lattice points placed in a very specific manner. From this description we conclude that all the empty tetrahedra in a non-spanning $3$-polytope $P$ have the same volume and they form a triangulation of $P$, and we compute the $h^*$-vectors of all non-spanning $3$-polytopes. We also show that all spanning $3$-polytopes contain a unimodular tetrahedron, except for two particular $3$-polytopes with five lattice points.