Most $(0,1)$-polytopes are not normal

Abstract

We prove that the proportion of $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes that are normal tends to zero at least at a double exponential rate as $d\to\infty$. As a consequence, the same holds for any of the following classes given by the type of triangulation possible: (a) quadratic, (b) flag unimodular, (c) regular unimodular, or (d) unimodular, among others. We classify the $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes for $d\leq5$ according to whether they admit a unimodular, flag unimodular, or quadratic triangulation. In dimension five, exactly $175$ out of $1{,}226{,}525$ classes have a flag unimodular triangulation, but no quadratic triangulation. Among them, there are polytopes whose toric rings are not Koszul; thus, we find the first polytopes that have a flag unimodular triangulation, but whose toric ring is not Koszul. In contrast with the matroid case, we exhibit a delta-matroid polytope that is not normal.