Toric Fano -folds
A toric Fano
-fold of index
is given by a unique smooth Gorenstein polytope
of index
.
The following table (corresponding to Corollary 2.2) shows how many manifolds exist for the given values.
At the end of each column is a download link for a package containing the corresponding smooth Gorenstein polytopes of index
in
polymake format.
|
|
-1 | 0 | 1 | 2 | 3 |
4 | 5 | 6 | 7 |
|
2 |
5 | 1 | 1 | 3 | | |
| | | |
|
3 |
18 | 1 | | 2 | 15 | |
| | | |
|
4 |
124 | 1 | | 1 | 4 |
118 | | | | |
|
5 |
866 | 1 | | | 1 |
11 | 853 | | | |
|
6 |
7622 | 1 | | | 1 |
3 | 27 | 7590 | | |
|
7 |
72256 | 1 | | | |
1 | 4 | 83 | 72167 | |
|
8 |
749892 | 1 | | | |
1 | 2 | 12 | 256 | 749620 |
|
9 |
8229721 | 1 | | | |
| 1 | 4 | 23 | 891 |
|
10 |
? | 1 | | | |
| 1 | 2 | 6 | 63 |
|
11 |
? | 1 | | | |
| | 1 | 3 | 13 |
|
12 |
? | 1 | | | |
| | 1 | 2 | 6 |
|
13 |
? | 1 | | | |
| | | 1 | 3 |
|
14 |
? | 1 | | | |
| | | 1 | 2 |
|
15 |
? | 1 | | | |
| | | | 1 |
|
16 |
? | 1 | | | |
| | | | 1 |
|
|
|
|
tgz |
tgz |
tgz |
tgz |
tgz |
tgz |
tgz |
tgz |
all |
Notes:
The small red numbers in the first column are the total number of reflexive polytopes in a given dimension. They are available from the
database of Andreas Paffenholz.
The small red numbers to the top-right, corresponding to
,
are reflexive polytopes which are not divisible. They are not contained in the packages for size reasons.
Smooth Gorenstein polytopes
The following packages contain all smooth Gorenstein polytopes of dimension
and index
we could compute so far.
Note:
Contrary to the first bunch of packages, the focus in this table is on polytopes. Hence, these packages include different polytopes which define the same toric Fano manifold. For example, dim03.tgz
contains both , a Gorenstein polytope of index 4, and , a Gorenstein polytope of index 2, as separate files.
We computed the stringy Hodge pairs
of all such polytopes with Calabi-Yau dimension
.
Seven pairs were not listed in the
Calabi-Yau manifold explorer by Benjamin Jurke:
(84,0), (85,1), (52,2), (69,1), (65,1), (55,2), (63,2)
The first five of these pairs do already appear in the
database of reflexive Gorenstein cones by Maximilian Kreuzer and Harald Skarke.
Below is a table of the corresponding smooth Gorenstein polytopes together with the stringy E-polynomials of the dual Gorenstein polytopes. The polytopes are available for download and given by the coordinate inequalities
together with the additional ones listed in each row.
Polytope |
| Stringy E-polynomial, Additional inequalities |
|
6, 2
|
|
|
6, 2
|
|
|
8, 3
|
|
|
8, 3
|
|
|
8, 3
|
|
|
8, 3
|
|
|
10, 4
|
|
|
12, 5
|
|
|
12, 5
|
|