Non-spanning lattice 3-polytopes

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 $h^{\ast }$-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.

20 pages. Changes from v2: small changes requested by journal referee; corrected typos in Thm 1.3; updated references

References in corpus (4)

• Ehrhart Theory of Spanning Lattice Polytopes
• Lattice 3-polytopes with six lattice points
• Enumeration of lattice 3-polytopes by their number of lattice points
• Spanning Lattice Polytopes and the Uniform Position Principle

Cited by in corpus (2)

• Elementary moves on lattice polytopes
• The covering radius and a discrete surface area for non-hollow simplices