PD Dr. Andreas Paffenholz

Regular Unimodular Triangulations

This page collects examples discussed in the survey "Existence of Unimodular Triangulations --- Positive Results" by Christian Haase, Andreas Paffenholz, Lindsay Piechnik, and Francisco Santos (see arxiv:1405.1687)

Regular unimodular triangulations of smooth reflexive polytopes

Summary of our results

As explained in the paper we can use a project-and-lift approach to check that a polytope has a regular unimodular triangulation (however, we cannot obtain negative results, if the method fails the polytope may still have a regular unimodular triangulation).

The following table summarizes the current status of the check for a regular unimodular triangulation (RUT) smooth reflexive polytopes.
Dimension 3 4 5 6 7 8 9
no. of polytopes 18 124 866 7622 72256 749892 8229721
RUT 18 124 866 7622 72256 checking checking
quadratic 18 124 866 ≥7620 ≥72240 checking checking
facet unimodular 18 96 554 4097 31881 checking checking

In the following we provide files that contain all polytopes where we haven't found a projection down to dimension one, sorted by the lowest dimension we found a projection for. We provide data for those polytopes in polymake format.

Polytopes without projection to dimension one

The computations can be checked with polymake. For the projections you need the extension Push-Forward Projections. The data above can be checked with the following commands (we assume you have loaded one of the polyhedral complexes above into a variable $p in the polymake shell, and you have switched to application fan)