Lattice 3-Polytopes with Few Lattice Points

This paper is intended as a first step in a program for a full algorithmic enumeration of lattice 3-polytopes. The program is based in the following two facts, that we prove: • For each n there is only a finite number of (equivalence classes of) 3polytopes of lattice width larger than one, where n is the number of lattice points. Polytopes of width one are infinitely many, but easy to classify. • There is only a finite number of 3-polytopes of lattice width larger than one that cannot be obtained by either glueing two smaller ones or by lifting in a very specific manner one of a list of five lattice 2-polytopes. The 3-polytopes in this finite list have at most 11 lattice points. For n = 4, all empty tetrahedra have width one (White). For n = 5 we here show that there are exactly 9 different 3-polytopes of width 2, and none of larger width. Eight of them are the clean tetrahedra previously classified by Kasprzyk and (independently) Reznick. In a subsequent paper, we show that for n = 6 there are 74 polytopes of width 2, two of width 3, and none of larger width.