Smooth Gorenstein polytopes

This page contains the polytopes described in On Smooth Gorenstein Polytopes by Benjamin Nill and Benjamin Lorenz.

Toric Fano d-folds

A toric Fano d-fold of index iX is given by a unique smooth Gorenstein polytope P of index r=iX.
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 iX>1 in polymake format.
-10123 4567 d-iX
2 5113
3 181215
4 124114 118
5 86611 11853
6 762211 3277590
7 722561 148372167
8 7498921 1212256749620
9 82297211 1423891
10 ?1 12663
11 ?1 1313
12 ?1 126
13 ?1 13
14 ?1 12
15 ?1 1
16 ?1 1
d 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 iX=1, 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 d and index r we could compute so far.
d 12345 678910 1112
r 2 345
tgz tgz tgz tgz tgz tgz tgz tgz tgz tgz tgz tgz
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 S3, a Gorenstein polytope of index 4, and 2S3, a Gorenstein polytope of index 2, as separate files.

Stringy Hodge numbers (Remark 5.5)

We computed the stringy Hodge pairs h 1 , 1 h 2 , 1 of all such polytopes with Calabi-Yau dimension n = d + 1 - 2 r = 3 . 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 x i 0 together with the additional ones listed in each row.
Polytope d , r Stringy E-polynomial,
Additional inequalities
P 0 = 2 S 3 × 2 S 3 6, 2 E s t P 0 × ; u v = u 3 v 3 + 69 u 2 v 2 - u 3 - u 2 v - u v 2 - v 3 + 69 u v + 1
i = 1 3 x i 2 i = 4 6 x i 2
P 1 6, 2 E s t P 1 × ; u v = u 3 v 3 + 85 u 2 v 2 - u 3 - u 2 v - u v 2 - v 3 + 85 u v + 1
i = 1 3 x i 2 -2x1+ i = 4 6 x i 1
P 2 = 3 S 8 8, 3 E s t P 2 × ; u v = u 3 v 3 + 84 u 2 v 2 - u 3 - v 3 + 84 u v + 1
i = 1 8 x i 3
P 3 8, 3 E s t P 3 × ; u v = u 3 v 3 + 69 u 2 v 2 - u 3 - u 2 v - u v 2 - v 3 + 69 u v + 1
i = 1 5 x i 2 -x1+ i = 6 8 x i 1
P 4 8, 3 E s t P 4 × ; u v = u 3 v 3 + 55 u 2 v 2 - u 3 - 2 u 2 v - 2 u v 2 - v 3 + 55 u v + 1
x 1 + x 2 1 -x1+ i = 3 5 x i 1 -x2+ i = 6 8 x i 1
P 5 8, 3 E s t P 5 × ; u v = u 3 v 3 + 63 u 2 v 2 - u 3 - 2 u 2 v - 2 u v 2 - v 3 + 63 u v + 1
x 1 + x 2 1 -x1+ i = 3 5 x i 1 -x3+ i = 6 8 x i 1
P 6 = S 3 × 2 S 7 10, 4 E s t P 6 × ; u v = u 3 v 3 + 65 u 2 v 2 - u 3 - u 2 v - u v 2 - v 3 + 65 u v + 1
i = 1 3 x i 1 i = 4 10 x i 2
P 7 12, 5 E s t P 7 × ; u v = u 3 v 3 + 65 u 2 v 2 - u 3 - u 2 v - u v 2 - v 3 + 65 u v + 1
i = 1 4 x i 1 i = 5 12 x i - i = 1 4 x i 1
P 8 = S 4 × S 4 × S 4 12, 5 E s t P 8 × ; u v = u 3 v 3 + 52 u 2 v 2 - u 3 - 2 u 2 v - 2 u v 2 - v 3 + 52 u v + 1
i = 1 4 x i 1 i = 5 8 x i 1 i = 9 12 x i 1