On the enumeration of lattice $3$-polytopes

Mónica Blanco, Francisco Santos

Published 2016-01-11Version 1

A lattice $3$-polytope is a polytope $P$ with integer vertices. We call size of $P$ the number of lattice points it contains, and width of $P$ the minimum, over all integer linear functionals $f$, of the length of the interval $f(P)$. In a previous paper we have shown that all but finitely many lattice $3$-polytopes of a given size have width one, which opens the possibility of enumerating those of width larger than one. There is none of size 4 (White, 1964) and those of sizes five and six were classified in our previous papers. In this paper we prove that every lattice $3$-polytope $P$ of width larger than one and size at least seven falls into one of the following three categories: - It projects in a very specific manner to one of a list of seven particular $2$-polytopes. We call $3$-polytopes of this type \emph{spiked}, and we describe them explicitly. - All except three of the lattice points in $P$ are contained in a rational parallelepiped of width one with respect to every facet. We call these polytopes \emph{boxed}. They have size at most 11 and we have completely enumerated them. - $P$ has (at least) two vertices $u$ and $v$ such that, when removing each of them, the width is still larger than one. Polytopes of this type can be all obtained "gluing" smaller polytopes of width larger than one. This allows for a computational enumeration of all lattice $3$-polytopes of width larger than one up to any given size. We have completely enumerated $3$-polytopes of width larger than one and size up to eleven. In particular, this implies the complete classification of "distinct-pair-sum" (or dps, for short) $3$-polytopes, which have size at most eight, of width larger than one.