Mathematics Colloquia and Seminars

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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.