Unimodular triangulations of dilated 3-polytopes

Abstract:

A seminal result in the theory of toric varieties, by Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds, showing that:

1. It contains all composite numbers.

2. It is an additive semigroup.

These two properties imply that the only values of $k$ that may not work (besides $1$ and $2$, which are known not to work) are $k\in \{3,5,7,11\}$. With an ad-hoc construction we show that $k=7$ and $k=11$ also work, except in this case the triangulation cannot be guaranteed to be “standard” in the boundary. All in all, the only open cases are $k=3$ and $k=5$.