Lattice 3-polytopes with six lattice points

TL;DR

This paper classifies all 3-dimensional lattice polytopes with exactly six lattice points, detailing their widths, invariants, and interior points, providing explicit coordinates and a comprehensive overview of their geometric properties.

Contribution

It provides the first complete classification of 3-polytopes with six lattice points, including explicit coordinates, invariants, and a detailed analysis of their widths and interior points.

Findings

74 polytopes of width 2

2 polytopes of width 3

None of larger width

Abstract

We classify lattice $3$-polytopes of width larger than one and with exactly $6$ lattice points. We show that there are $74$ polytopes of width $2$, two polytopes of width $3$, and none of larger width. We give explicit coordinates for representatives of each class, together with other invariants such as their oriented matroid (or order type) and volume vector. For example, according to the number of interior points these $76$ polytopes divide into $23$ tetrahedra with two interior points (clean tetrahedra), $49$ polytopes with one interior point (the $49$ canonical three-polytopes with five boundary points previously classified by Kasprzyk) and only $4$ hollow polytopes. We also give a complete classification of three-polytopes of width one with $6$ lattice points. In terms of the oriented matroid of these six points, they lie in eight infinite classes and twelve individual polytopes. Our motivation comes partly from the concept of distinct pair sum (or dps) polytopes, which, in dimension $3$, can have at most $8$ lattice points. Among the $74+2$ classes mentioned above, exactly $44 + 1$ are dps.