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

In dimension six there are 2 such polytopes (numbers 803 and 2616 in the lists given in the polymake database ). For the first we found a projection to dimension 4, for the second we found no projection. We provide them as an array of polytopes for polymake (to be loaded with load_data):

fano-6d-not_fully_projected.pdata

Both polytopes are facet unimodular, so they do have a regular unimodular triangulation.
In dimension seven there are 16 such polytopes. We provide them as an array of polytopes for polymake (to be loaded with load_data)
- Lowest dimension 2: fano-7d-not_fully_projected_dim2.pdata
- Lowest dimension 3: fano-7d-not_fully_projected_dim3.pdata
- Lowest dimension 4: fano-7d-not_fully_projected_dim4.pdata
- Lowest dimension 6: fano-7d-not_fully_projected_dim6.pdata
In total these files contain 16 polytopes. All other 7-dimensional polytopes project into dimension one and thus have a quadratic triangulation.

For the remaining we can check whether at least we have a regular unimodular triangulation. This is clear for those that project to dimension two. We can check that out of the remaining eleven polytopes in dimensions three and above all but three are facet unimodular. The three exceptions are the polyopes no. 4065, 5079, and 26254 in the polymake database , whose lowest projection dimension we were able to find is three.

Direct inspection shows that all three polytopes each have regular unimodular triangulations that respect the subdivsion hyperplanes obtained from the projections. We provide the relevant data to check this again in polymake format. The following files contain objects of type PolyhedralComplex. You should switch to application fan prior to loading them.
- 4065: unimodular_triangulation_of_4065_complete.pcom
  The polytope is subdivided into four parts by the hyperplane subdivision obtained from the projection. Here are the four parts separately with their projection:
- 5079: unimodular_triangulation_of_4065_complete.pcom
  The polytope is subdivided into two parts by the hyperplane subdivision obtained from the projection. Here are the four parts separately with their projection:
  - unimodular_triangulation_of_5079_cell0.pcom
  - unimodular_triangulation_of_5079_cell1.pcom
- 26254: unimodular_triangulation_of_4065_complete.pcom
  The polytope has no subdivision hyperplanes induced by the projection.

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)

regularity:
fan > is_regular($p->VERTICES,$p->MAXIMAL_POLYTOPES);
proper triangulation: ,
fan > check_fan($p->VERTICES,$p->MAXIMAL_POLYTOPES);
unimodularity:
fan > $q=new Polytope(POINTS=>$p->VERTICES); print $p->N_MAXIMAL_POLYTOPES==$q->LATTICE_VOLUME;