This paper is a joint paper of Mathieu Dutour-Sikirić, Alexey Garber, and Alexander Magazinov.

In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all five-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in R⁵ holds if and only if every five-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron P is combinatorially Voronoi, we mean that P is combinatorially equivalent to a Dirichlet-Voronoi polytope for some lattice Λ, and this cominatorial equivalence is naturally translated into equivalence of the tiling by copies of P with the Voronoi tiling of Λ.

We also propose a new sufficient condition implying that a parallelohedron is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex.

The paper is available at arXiv.

The related source code and data files are available under supporting files tab below.

Five-dimensional Dirichlet-Voronoi parallelohedra

Information about face lattices of five-dimensional Dirichlet-Voronoi parallelohedra is contained in the following files. The parallelohedra are grouped based on the number extreme rays defining the corresponding secondary cone.

• FaceLatt_5_1.txt
• FaceLatt_5_2.txt
• FaceLatt_5_3.txt
• FaceLatt_5_4.txt
• FaceLatt_5_5.txt
• FaceLatt_5_6.txt
• FaceLatt_5_7.txt
• FaceLatt_5_8.txt
• FaceLatt_5_9.txt
• FaceLatt_5_10.txt
• FaceLatt_5_11.txt
• FaceLatt_5_12.txt
• FaceLatt_5_13.txt
• FaceLatt_5_14.txt
• FaceLatt_5_15.txt

Dual 3-cells

Information about dual 3-cells of five-dimensional Dirichlet-Voronoi parallelohedra is contained in the following files. The parallelohedra are grouped based on the number extreme rays defining the corresponding secondary cone
• DualCell_data_01.txt
• DualCell_data_02.txt
• DualCell_data_03.txt
• DualCell_data_04.txt
• DualCell_data_05.txt
• DualCell_data_06.txt
• DualCell_data_07.txt
• DualCell_data_08.txt
• DualCell_data_09.txt
• DualCell_data_10.txt
• DualCell_data_11.txt
• DualCell_data_12.txt
• DualCell_data_13.txt
• DualCell_data_14.txt
• DualCell_data_15.txt

Source code

• Python scripts that transform face lattice data into dual cell data and generate a GAP program to work with the Venkov complex based on dual cell data.
  • Python script for face lattice data;
  • Python script for dual cell data;
  • Common readme for both scripts.
• SAGE script that checks GGM condition based on dual cell data. You might need to change the filenames in the beginning of the script based on their location.