Enumerating lattice 3-polytopes

Abstract

A lattice 3-polytope P⊂R3 is the convex hull of finitely many points from Z3. The size of P is the number of integer points it contains, and the width of P is the minimum, over all integer linear functionals f, of the length of the interval f(P ).We present our results on a full enumeration of lattice 3-polytopes via their size and width: for any fixed n≥d+1 there are infinitely many 3-polytopes of width one and size n, but they are easy to describe (they lie between two consecutive lattice planes). Those of width larger than one are finitely many for each size, and the full list of them can be obtained by one of two methods: (a) Most of them contain two proper subpolytopes of width larger than one, and thus can be obtained from the list of size n−1 using computer algorithms. (b) The rest have very precise structural properties that allow for a direct enumeration of them.We have implemented the algorithms in MATLAB and run it until obtaining the following: There are 9, 76, 496, 2675, 11698, 45035 and 156464 lattice 3-polytopes of width larger than one and of sizes 5, 6, 7, 8, 9, 10 and 11, respectively.