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

Centrally symmetric 2-level polytopes in dimension 6

f-vector # faces # vertices + # facets Mahler volume
(12,60,160,240,192,64)730765.68888888888889
(14,72,182,244,168,48)730625.68888888888889
(14,84,240,330,200,40)910545.86666666666667
(16,100,270,334,180,32)934485.89037037037037
(16,82,196,242,152,40)730565.68888888888889
(16,88,204,240,144,36)730525.68888888888889
(16,88,222,276,162,36)802525.76
(18,102,244,280,150,30)826485.7837037037037
(18,108,272,312,158,28)898465.85481481481481
(18,88,200,240,146,36)730545.68888888888889
(18,96,226,260,144,32)778505.7362962962963
(20,100,216,232,128,32)730525.68888888888889
(20,106,238,262,138,28)794485.7520987654321
(20,108,246,264,134,28)802485.76
(20,114,264,284,140,26)850465.80740740740741
(20,120,290,310,144,24)910445.86666666666667
(20,90,200,240,144,34)730545.68888888888889
(22,106,220,230,122,28)730505.68888888888889
(22,122,270,278,132,24)850465.80740740740741
(22,128,282,284,132,24)874465.83111111111111
(22,130,300,300,132,24)910465.86666666666667
(24,108,220,230,120,26)730505.68888888888889
(24,116,232,232,116,24)746485.70469135802469

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.