PD Dr. Andreas Paffenholz

Smooth Reflexive Lattice Polytopes

The following lists contain all smooth reflexive d-polytopes for d between 3 and 9 up to lattice equivalence. The lists were computed with the algorithm of Øbro (see arxiv: 0704.0049). Oebro used his algorithm to compute the lists up to dimension 8. We wrote an improved implementation of his algorithm within the polymake framework that allowed us to compute the list in dimension 9. Computations of certain higher dimensional classes are in preparation.

For dimensions 3 to 8 we have provide the data in polymake format. The tarballs can directly be read into polymake using the tarball-script described here. Each polytope only contains the list of vertices. Other properties can be computed with the lattice polytope extension of polymake that we have written (see the lattice polytope tutorial).
The 9-dimensional data is too large to provide it in polymake format. We give plain text files that contain lists of facets. We also have it in polymake format, so contact us if you need it.

We have already computed various of their properties. A table with some more information about low dimensional smooth reflexive polytopes is in preparation. Contact us if you have a particular question.


Dimension 3 4 5 6 7 8 9
no. of polytopes 18 124 866 7622 72256 749892 8229721

All smooth reflexive polytopes in dimension ≤8 are normal, and most of them have a regular unimodular triangulation (Haase and Paffenholz: On Fanos and Chimneys).

Lists of Polytopes