Experimental results on 2-level polytopes
This page reports experimental results and data about the enumeration of 2-level polytopes (a.k.a. compressed polytopes).
See here for the latest data.
A complete list (in polymake format) of computed 2-level polytopes up to isomorphism and up to dimension 7:
| Dimension | List of polytopes |
|---|---|
| 3 | two_level_poly_3.tgz |
| 4 | two_level_poly_4.tgz |
| 5 | two_level_poly_5.tgz |
| 6 | two_level_poly_6.tgz |
| 7 | two_level_poly_7.tgz |
The current state-of-the-art regarding the number of 2-level polytopes and interesting classes is depicted in the following table. More data about special cases of 2-level polytopes follow.
Number of 2-level polytopes and classes
| Dimension | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| 2-level polytopes | 5 | 19 | 106 | 1150 | 27292 | |
| 2-level suspensions | 4 | 15 | 88 | 956 | 23279 | |
| polar 2-level polytopes | 4 | 12 | 42 | 276 | ||
| simplicial facet | 4 | 12 | 41 | 248 | ||
| Stable sets polytopes of perfect graphs | 4 | 11 | 33 | 148 | 906 | 8887 |
| centrally symmetric | 2 | 4 | 13 | 45 | 239 | |
| Hanner polytopes | 2 | 4 | 8 | 18 | 40 | |
| faces of Birkhoff | 4 | 11 | 33 | 129 | 661 | 4530 |
| simplicial | 2 | 3 | 2 | 4 | 2 | |
| 0/1 | 8 | 192 | 1048576 |
- Stable sets of perfect graphs taken from [S. Hougardy, Classes of perfect graphs, Discrete Mathematics no.19-20, 2529-2571, 2006].
- Polar 2-level are 2-level polytopes whose polar is 2-level
- Birkhoff polytope faces taken from here.
Centrally symmetric 2-level polytopes in dimension 6
| f-vector | # faces | # vertices + # facets | Mahler volume |
|---|---|---|---|
| (12,60,160,240,192,64) | 730 | 76 | 5.68888888888889 |
| (14,72,182,244,168,48) | 730 | 62 | 5.68888888888889 |
| (14,84,240,330,200,40) | 910 | 54 | 5.86666666666667 |
| (16,100,270,334,180,32) | 934 | 48 | 5.89037037037037 |
| (16,82,196,242,152,40) | 730 | 56 | 5.68888888888889 |
| (16,88,204,240,144,36) | 730 | 52 | 5.68888888888889 |
| (16,88,222,276,162,36) | 802 | 52 | 5.76 |
| (18,102,244,280,150,30) | 826 | 48 | 5.7837037037037 |
| (18,108,272,312,158,28) | 898 | 46 | 5.85481481481481 |
| (18,88,200,240,146,36) | 730 | 54 | 5.68888888888889 |
| (18,96,226,260,144,32) | 778 | 50 | 5.7362962962963 |
| (20,100,216,232,128,32) | 730 | 52 | 5.68888888888889 |
| (20,106,238,262,138,28) | 794 | 48 | 5.7520987654321 |
| (20,108,246,264,134,28) | 802 | 48 | 5.76 |
| (20,114,264,284,140,26) | 850 | 46 | 5.80740740740741 |
| (20,120,290,310,144,24) | 910 | 44 | 5.86666666666667 |
| (20,90,200,240,144,34) | 730 | 54 | 5.68888888888889 |
| (22,106,220,230,122,28) | 730 | 50 | 5.68888888888889 |
| (22,122,270,278,132,24) | 850 | 46 | 5.80740740740741 |
| (22,128,282,284,132,24) | 874 | 46 | 5.83111111111111 |
| (22,130,300,300,132,24) | 910 | 46 | 5.86666666666667 |
| (24,108,220,230,120,26) | 730 | 50 | 5.68888888888889 |
| (24,116,232,232,116,24) | 746 | 48 | 5.70469135802469 |
List of 4-dimensional 2-level polytopes with their properties
| operation | f-vector | face pairs | STAB | simplicial-facet | polar |
|---|---|---|---|---|---|
| pyramid over simplex | (5,10,10,5) | (4,6,4) -- (1) | X | X | X |
| pyramid over square base pyramid | (6,13,13,6) | (4,6,4) -- (2)(5,8,5) -- (1) | X | X | X |
| Birkhoff | (6,15,18,9) | (4,6,4) -- (2) | X | X | |
| pyramid over triangular prism | (7,15,14,6) | (4,6,4) -- (3,3) (5,8,5) -- (2)(6,9,5) -- (1) | X | X | |
| dual wedge over edge of square base pyr. | (7,17,18,8) | (4,6,4) -- (3,3) (5,8,5) -- (2) | X | X | X |
| pyramid over cross-polytope | (7,18,20,9) | (4,6,4) -- (3,3) (6,12,8) -- (1) | X | X | |
| prism over simplex | (8,16,14,6) | (4,6,4) -- (4,6,4) (6,9,5) -- (2) | X | X | |
| wedge over edge of square base pyramid | (8,18,17,7) | (4,6,4) -- (4,4) (5,8,5) -- (3,3) (6,9,5) -- (2) | X | X | X |
| unknown | (8,21,22,9) | (4,6,4) -- (4,6,4) (5,8,5) -- (3,3) (6,12,8) -- (2) | X | ||
| bipyramid over cross-polytope | (8,24,32,16) | (4,6,4) -- (4,6,4) | X | X* | |
| wedge over triangle of triangular prism; dual Birkhoff | (9,18,15,6) | (6,9,5) -- (3,3) | X | X | |
| pyramid over cube | (9,20,18,7) | (5,8,5) -- (4,4) (8,12,6) -- (1) | X | X | |
| wedge over facet of cross-polytope | (9,24,24,9) | (4,6,4) -- (5,8,5) (6,12,8) -- (3,3) (6,9,5) -- (3,3) | X | ||
| prism over square-base pyramid | (10,21,18,7) | (5,8,5) -- (5,8,5) (6,9,5) -- (4,4) (8,12,6) -- (2) | X | ||
| bipyramid over cube | (10,28,30,12) | (5,8,5) -- (5,8,5) | X* | ||
| hypersimplex | (10,30,30,10) | (4,6,4) -- (6,12,8) | X | ||
| prism over triangular prism product of triangle, square | (12,24,19,7) | (6,9,5) -- (6,9,5) (8,12,6) -- (4,4) | X | ||
| prism over cross-polytope | (12,30,28,10) | (6,9,5) -- (6,9,5) (6,12,8) -- (6,12,8) | X* | ||
| prism over cube | (16,32,24,8) | (8,12,6) -- (8,12,6) | X | X* |
X* means that the polytope is centrally symmetric.