Unimodular covers of 3-dimensional parallelepipeds and Cayley sums

TL;DR

This paper proves that certain classes of 3D lattice polytopes, including parallelepipeds, centrally symmetric polytopes, and specific Cayley sums, have unimodular covers, advancing understanding of their lattice properties.

Contribution

It establishes unimodular covers for these classes of 3D lattice polytopes, extending previous results that only showed the IDP property.

Findings

Parallelepipeds in 3D have unimodular covers.

Centrally symmetric 3D polytopes have unimodular covers.

Cayley sums with refined normal fans have unimodular covers.

Abstract

We show that the following classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds, the class of centrally symmetric polytopes, and the class of Cayley sums $\text{Cay}(P,Q)$ where the normal fan of $Q$ refines that of $P$. This improves results of Beck et al.~(2018) and Haase et al.~(2008) where the last two classes were shown to be IDP.