# Mathematics Colloquia and Seminars

### Enumerating lattice $3$-polytopes
A lattice 3$-polytope$P \subset \mathbb{R}^3$is the convex hull of finitely many integer points (points in$\Z^3$). We call size of$P$the number of integer points it contains, and width of$P$the minimum, over all integer linear functionals$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$n \ge d+1$there are only finitely many lattice$3$-polytopes of width larger than one and size$n$. Lattice$3$-polytopes of width one are infinitely many but easy to describe. The polytopes in the finite list of width larger than one and each fixed size$n$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 ones that cannot have very precise structural properties that allow for a direct enumeration of them. We have implemented the algorithms in MATLAB and run the algorithm until obtaining a complete list of lattice$3$-polytopes of width larger than one and size up to eleven. There are$9$,$76$,$496$,$2675$,$11698$,$45035$and$156464$of sizes$5$,$6$,$7$,$8$,$9$,$10$and$11\$, respectively.