Unimodular triangulations of dilated 3-polytopes

TL;DR

This paper investigates which dilations of 3-dimensional lattice polytopes admit unimodular triangulations, showing most values of k work except for a few small primes, with some remaining open cases.

Contribution

It characterizes the set of dilation factors k for which unimodular triangulations exist in 3-polytopes, identifying nearly all such values except for 3 and 5.

Findings

All composite numbers k work.

The set of k values forms an additive semigroup.

Only k=3 and k=5 remain unresolved.

Abstract

A seminal result in the theory of toric varieties, due to 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 $k=5$.