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 $i_X$ is given by a unique smooth Gorenstein polytope $P$ of index $r=i_X$.
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 $i_X>1$ in polymake format.

		-1	0	1	2	3	4	5	6	7	$d-i_X$
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
$d$			tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz	all

Smooth Gorenstein polytopes

The following packages contain all smooth Gorenstein polytopes of dimension $d$ and index $r$ we could compute so far.

$d$	1	2	3	4	5	6	7	8	9	10	11	12
$r\ge$	2									3	4	5
	tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz	tgz

We computed the stringy Hodge pairs $\left(h^{1,1},h^{2,1}\right)$ of all such polytopes with Calabi-Yau dimension $n=d+1-2r=3$. 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 $x_i\ge 0$ together with the additional ones listed in each row.

Polytope	$d,r$	Stringy E-polynomial, Additional inequalities
$P_0=2S_3\times 2S_3$	6, 2	$E_{st}\left(P_0^{\times };u,v\right)=u^3{v}^3+69u^2{v}^2-u^3-u^{2}v-u{v}^2-v^3+69uv+1$ $\sum _{i=1}^3{x}_i\le 2\sum _{i=4}^6{x}_i\le 2$
$P_1$	6, 2	$E_{st}\left(P_1^{\times };u,v\right)=u^3{v}^3+85u^2{v}^2-u^3-u^{2}v-u{v}^2-v^3+85uv+1$ $\sum _{i=1}^3{x}_i\le 2-2x_1+\sum _{i=4}^6{x}_i\le 1$
$P_2=3S_8$	8, 3	$E_{st}\left(P_2^{\times };u,v\right)=u^3{v}^3+84u^2{v}^2-u^3-v^3+84uv+1$ $\sum _{i=1}^8{x}_i\le 3$
$P_3$	8, 3	$E_{st}\left(P_3^{\times };u,v\right)=u^3{v}^3+69u^2{v}^2-u^3-u^{2}v-u{v}^2-v^3+69uv+1$ $\sum _{i=1}^5{x}_i\le 2-x_1+\sum _{i=6}^8{x}_i\le 1$
$P_4$	8, 3	$E_{st}\left(P_4^{\times };u,v\right)=u^3{v}^3+55u^2{v}^2-u^3-2u^{2}v-2u{v}^2-v^3+55uv+1$ $x_1+x_2\le 1-x_1+\sum _{i=3}^5{x}_i\le 1-x_2+\sum _{i=6}^8{x}_i\le 1$
$P_5$	8, 3	$E_{st}\left(P_5^{\times };u,v\right)=u^3{v}^3+63u^2{v}^2-u^3-2u^{2}v-2u{v}^2-v^3+63uv+1$ $x_1+x_2\le 1-x_1+\sum _{i=3}^5{x}_i\le 1-x_3+\sum _{i=6}^8{x}_i\le 1$
$P_6=S_3\times 2S_7$	10, 4	$E_{st}\left(P_6^{\times };u,v\right)=u^3{v}^3+65u^2{v}^2-u^3-u^{2}v-u{v}^2-v^3+65uv+1$ $\sum _{i=1}^3{x}_i\le 1\sum _{i=4}^{10}{x}_i\le 2$
$P_7$	12, 5	$E_{st}\left(P_7^{\times };u,v\right)=u^3{v}^3+65u^2{v}^2-u^3-u^{2}v-u{v}^2-v^3+65uv+1$ $\sum _{i=1}^4{x}_i\le 1\sum _{i=5}^{12}{x}_i-\sum _{i=1}^4{x}_i\le 1$
$P_8=S_4\times S_4\times S_4$	12, 5	$E_{st}\left(P_8^{\times };u,v\right)=u^3{v}^3+52u^2{v}^2-u^3-2u^{2}v-2u{v}^2-v^3+52uv+1$ $\sum _{i=1}^4{x}_i\le 1\sum _{i=5}^8{x}_i\le 1\sum _{i=9}^{12}{x}_i\le 1$