\documentclass[10pt,a4paper]{article}

\input{Definitions}

\begin{document}

\section{Polyhedron $ 2 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&0&0&1&0&0&1&1&1&1&1
\\\noalign{\medskip}0&1&1&1&0&1&1&0&0&0&1&1\\\noalign{\medskip}0&0&0&-
1&1&0&0&0&-1&-1&-1&0\\\noalign{\medskip}0&0&0&-2&0&0&1&0&1&0&-1&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0&0&
-1&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&-1&-1&0&0&0&-1&0&0&1&1
\\\noalign{\medskip}-2&0&0&0&1&1&-1&0&1&0&0&0\\\noalign{\medskip}0&-2&0
&0&0&1&0&0&1&-2&1&0\\\noalign{\medskip}0&0&1&1&1&0&0&1&0&0&0&-2
\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&1&-2&0\\\noalign{\medskip}0&0&0&0
&0&-1&1&0&-1&1&0&0\end {array} \right] 
\end{equation}
This Mori cone is not simplicial. 
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{4}}+D_{{5}}
$, $K_{{1}}
=1/2\,D_{{5}}+1/2\,D_{{8}}+D_{{4}}+1/2\,D_{{3}}-1/2\,D_{{1}}
$, $K_{{3}}
=1/2\,D_{{1}}+1/2\,D_{{3}}-1/2\,D_{{5}}-1/2\,D_{{8}}-D_{{4}}
$, $K_{{4}}
=-1/2\,D_{{3}}+1/2\,D_{{5}}+1/2\,D_{{8}}+D_{{4}}+1/2\,D_{{1}}
$, $K_{{5}}
=1/2\,D_{{8}}+D_{{4}}+3/2\,D_{{5}}+1/2\,D_{{3}}-1/2\,D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{5}}D_{{8}},D_{{5}}D_{{9}},D_{{5}}D_
{{10}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{3}}D_{{4}}D_{{7}},D_{{4}}D_
{{7}}D_{{9}},D_{{4}}D_{{7}}D_{{10}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_{{1
}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{6}}D_{{8}},D_{{1}}D_{{2}}D_{
{4}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 44 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&0&0&0&1&1&-1&0&1&0&0&0
\\\noalign{\medskip}-2&-4&1&1&1&0&2&1&0&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}-1/2\,K_{{2}}
\qquad
J_{{2}}=1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=5\,J_{{2}}J_{{1}}+6\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=32
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&832
&9048
&296256
&10896200
&496315712
\\ 
1 &32
&3968
&420544
&43292416
&4300353568
&417443075200
\\ 
2 &4
&5952
&2556784
&731520512
&159613720920
&29323275213056
\\ 
3 &0
&3968
&6550208
&4550144640
&1990567239584
&649617659336064
\\ 
4 &0
&832
&8831008
&14796791040
&12407439429504
&6902604520184960
\\ 
5 &0
&0
&6550208
&28878680064
&46220194846240
&42948676303436416
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&0&0&1&0&0&1&1&1&1&1
\\\noalign{\medskip}0&1&1&1&0&1&1&0&0&0&1&1\\\noalign{\medskip}0&0&0&-
1&1&0&0&0&-1&-1&-1&0\\\noalign{\medskip}0&0&0&-2&0&0&1&0&1&0&-1&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0&0&
-1&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&-1&-1&0&0&0&-1&0&0&1&1
\\\noalign{\medskip}0&-2&0&0&0&0&1&0&0&-1&1&0\\\noalign{\medskip}0&0&1
&1&1&0&0&1&0&0&0&-2\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&1&-2&0
\\\noalign{\medskip}0&0&0&0&0&1&-1&0&1&-1&0&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{4}}
=D_{{8}}+D_{{5}}+2\,D_{{4}}
$, $K_{{1}}
=-D_{{1}}+2\,D_{{4}}+D_{{3}}+D_{{8}}+D_{{5}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{5}}
=D_{{4}}
$, $K_{{2}}
=D_{{5}}+D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{4}}D_{{7}},D_{{5}}D_{{9}},D_{{5}}D_
{{10}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_
{{1}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{6}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 44 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&-4&1&1&1&0&2&1&0&0&0&0
\\\noalign{\medskip}-2&0&0&0&1&1&-1&0&1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{2}}
\qquad
J_{{2}}=1/2\,K_{{1}}-1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=2
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=32
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&32
&4
&0
&0
&0
\\ 
1 &832
&3968
&5952
&3968
&832
&0
\\ 
2 &9048
&420544
&2556784
&6550208
&8831008
&6550208
\\ 
3 &296256
&43292416
&731520512
&4550144640
&14796791040
&28878680064
\\ 
4 &10896200
&4300353568
&159613720920
&1990567239584
&12407439429504
&46220194846240
\\ 
5 &496315712
&417443075200
&29323275213056
&649617659336064
&6902604520184960
&42948676303436416
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 4 $}

\subsection{Partition $1$ of model $4$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} 1&0&1&0&1&0&1&0&0&0&1&1&1&1
\\\noalign{\medskip}0&1&0&1&0&1&0&1&1&1&1&1&1&1\\\noalign{\medskip}0&0
&0&-3&0&1&0&0&0&-1&-2&-2&-1&-1\\\noalign{\medskip}0&0&0&-1&1&0&0&-1&0&0
&-1&0&0&-1\\\noalign{\medskip}0&0&0&-2&0&0&0&1&1&0&-1&-1&-1&0
\\\noalign{\medskip}0&0&0&0&0&0&1&-1&0&0&0&0&0&0\\\noalign{\medskip}0&0
&1&-1&0&0&0&0&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} -1&0&0&0&0&0&0&0&1&-2&0&1&0&0
\\\noalign{\medskip}0&-2&0&0&0&1&-1&-1&1&0&0&0&0&1\\\noalign{\medskip}0
&-1&-1&-1&-1&0&0&0&0&0&0&1&1&0\\\noalign{\medskip}0&0&1&1&0&0&0&0&0&0&
-2&0&0&1\\\noalign{\medskip}0&0&1&1&1&0&0&0&0&1&0&-2&0&0
\\\noalign{\medskip}0&0&-1&-1&0&0&0&0&-1&1&1&0&0&0\\\noalign{\medskip}0
&0&1&1&1&1&0&0&0&0&0&0&-2&0\\\noalign{\medskip}0&0&0&0&0&0&1&1&0&0&1&0
&0&-2\end {array} \right] 
\end{equation}
This Mori cone is not simplicial. 
A positive basis for the generators of the K\"ahler cone is $K_{{7}}
=D_{{5}}
$, $K_{{6}}
=D_{{4}}
$, $K_{{3}}
=-D_{{1}}+D_{{3}}-D_{{9}}+D_{{5}}
$, $K_{{1}}
=D_{{7}}-D_{{9}}-2\,D_{{1}}+2\,D_{{3}}+D_{{5}}
$, $K_{{5}}
=-D_{{9}}-2\,D_{{1}}+2\,D_{{3}}+D_{{5}}
$, $K_{{2}}
=D_{{3}}+D_{{8}}+2\,D_{{7}}-D_{{9}}-2\,D_{{1}}+D_{{5}}+D_{{4}}
$, $K_{{4}}
=D_{{8}}+2\,D_{{7}}-D_{{9}}-2\,D_{{1}}+2\,D_{{3}}+D_{{5}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{9}},D_{{3}}D_{{12}},D_{{4}}D_{{8}},D_{{4}}D_{{10}},D_{{7}}
D_{{10}},D_{{7}}D_{{11}},D_{{8}}D_{{9}},D_{{8}}D_{{11}},D_{{8}}D_{{12}
},D_{{9}}D_{{10}},D_{{9}}D_{{11}},D_{{10}}D_{{11}},D_{{10}}D_{{12}},D_
{{11}}D_{{12}},D_{{1}}D_{{2}}D_{{12}},D_{{3}}D_{{5}}D_{{6}},D_{{4}}D_{
{7}}D_{{9}},D_{{4}}D_{{7}}D_{{12}},D_{{5}}D_{{6}}D_{{8}},D_{{5}}D_{{6}
}D_{{9}},D_{{5}}D_{{6}}D_{{10}},D_{{5}}D_{{6}}D_{{11}},D_{{1}}D_{{2}}D
_{{3}}D_{{4}},D_{{1}}D_{{2}}D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{3}}D_{{8}
},D_{{1}}D_{{2}}D_{{4}}D_{{7}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 44 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$, $D_{{12}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} -2&-6&1&1&1&3&0&0&2&0&0&0&0&0
\\\noalign{\medskip}-2&0&0&0&1&0&1&1&-1&0&0&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/6\,K_{{2}}
\qquad
J_{{2}}=-1/6\,K_{{2}}+1/2\,K_{{1}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=5\,J_{{2}}J_{{1}}+12\,{J_{{1}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=1
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=22
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&16
&2
&0
&0
&0
\\ 
1 &3648
&17088
&25632
&17088
&3648
&0
\\ 
2 &178680
&12354912
&77133912
&199319904
&269313408
&199319904
\\ 
3 &42027072
&8285397376
&150296908544
&957565747776
&3149354922240
&6180131665920
\\ 
4 &8753885256
&5317807400976
&221053760557740
&2877572824780752
&18316074821801472
&69043832283835152
\\ 
5 &2599593660480
&3310249645037376
&270328097125865088
&6392393185512636864
&70206444552330709632
&445275925022544774720
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} 1&0&1&0&1&0&1&0&0&0&1&1&1&1
\\\noalign{\medskip}0&1&0&1&0&1&0&1&1&1&1&1&1&1\\\noalign{\medskip}0&0
&0&-3&0&1&0&0&0&-1&-2&-2&-1&-1\\\noalign{\medskip}0&0&0&-1&1&0&0&-1&0&0
&-1&0&0&-1\\\noalign{\medskip}0&0&0&-2&0&0&0&1&1&0&-1&-1&-1&0
\\\noalign{\medskip}0&0&0&0&0&0&1&-1&0&0&0&0&0&0\\\noalign{\medskip}0&0
&1&-1&0&0&0&0&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} -2&0&0&0&1&0&1&1&-1&0&0&0&0&0
\\\noalign{\medskip}-1&0&-1&-1&0&0&0&0&0&-1&1&1&0&0
\\\noalign{\medskip}0&-1&-1&-1&-1&0&0&0&0&0&0&1&1&0
\\\noalign{\medskip}0&0&1&1&0&0&0&0&0&0&-2&0&0&1\\\noalign{\medskip}0&0
&1&1&1&0&0&0&0&1&0&-2&0&0\\\noalign{\medskip}0&0&1&1&1&1&0&0&0&0&0&0&-
2&0\\\noalign{\medskip}0&0&0&0&0&0&1&1&0&0&1&0&0&-2
\\\noalign{\medskip}0&0&0&0&0&0&-1&-1&1&-1&0&0&0&1\end {array}
 \right] 
\end{equation}
This Mori cone is not simplicial. 
A positive basis for the generators of the K\"ahler cone is $K_{{7}}
=D_{{7}}
$, $K_{{1}}
=-D_{{9}}-2\,D_{{1}}+D_{{7}}+2\,D_{{3}}+D_{{5}}
$, $K_{{2}}
=D_{{3}}-D_{{9}}+D_{{8}}-2\,D_{{1}}+2\,D_{{7}}+D_{{5}}+D_{{4}}
$, $K_{{4}}
=-D_{{9}}+D_{{8}}-2\,D_{{1}}+2\,D_{{7}}+2\,D_{{3}}+D_{{5}}
$, $K_{{3}}
=-D_{{1}}+D_{{7}}+D_{{3}}-D_{{9}}+D_{{5}}
$, $K_{{6}}
=D_{{5}}+D_{{7}}
$, $K_{{5}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{9}},D_{{3}}D_{{12}},D_{{4}}D_{{8}},D_{{4}}D_{{10}},D_{{7}}
D_{{9}},D_{{7}}D_{{10}},D_{{7}}D_{{11}},D_{{7}}D_{{12}},D_{{8}}D_{{11}
},D_{{9}}D_{{10}},D_{{9}}D_{{11}},D_{{10}}D_{{11}},D_{{10}}D_{{12}},D_
{{11}}D_{{12}},D_{{1}}D_{{2}}D_{{7}},D_{{1}}D_{{2}}D_{{12}},D_{{3}}D_{
{5}}D_{{6}},D_{{5}}D_{{6}}D_{{9}},D_{{5}}D_{{6}}D_{{10}},D_{{5}}D_{{6}
}D_{{11}},D_{{1}}D_{{2}}D_{{3}}D_{{4}},D_{{1}}D_{{2}}D_{{3}}D_{{8}},D_
{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 44 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$, $D_{{12}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccccc} -2&0&0&0&1&0&1&1&-1&0&0&0&0&0
\\\noalign{\medskip}-2&-6&1&1&1&3&0&0&2&0&0&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=-1/6\,K_{{2}}+1/2\,K_{{1}}
\qquad
J_{{2}}=1/6\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=5\,J_{{2}}J_{{1}}+12\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=2
&
K_{{2,2,2}}
&=1
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=22
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&3648
&178680
&42027072
&8753885256
&2599593660480
\\ 
1 &16
&17088
&12354912
&8285397376
&5317807400976
&3310249645037376
\\ 
2 &2
&25632
&77133912
&150296908544
&221053760557740
&270328097125865088
\\ 
3 &0
&17088
&199319904
&957565747776
&2877572824780752
&6392393185512636864
\\ 
4 &0
&3648
&269313408
&3149354922240
&18316074821801472
&70206444552330709632
\\ 
5 &0
&0
&199319904
&6180131665920
&69043832283835152
&445275925022544774720
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&1&1&1&0
\\\noalign{\medskip}0&1&1&1&1&0&0&0&1\\\noalign{\medskip}0&0&0&0&1&0&0
&0&-1\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -3&-2&1&1&0&1&1&1&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{1}}
$, $K_{{2}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
6\,K_{{2}}K_{{1}}+6\,{K_{{1}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 56 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -3&-2&1&1&0&1&1&1&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=4\,{J_{{1}}}^{2}+{J_{{2}}}^{2}+4\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=6
&
K_{{1,1,2}}
&=6
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=48
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&24
&0
&0
&0
&0
\\ 
1 &396
&1152
&396
&0
&0
&0
\\ 
2 &2610
&53136
&112068
&53136
&2610
&0
\\ 
3 &35640
&2377728
&15951564
&28024704
&15951564
&2377728
\\ 
4 &605844
&103323672
&1602730872
&6746381496
&10576809936
&6746381496
\\ 
5 &12212172
&4400303616
&132192153792
&1084701369600
&3472953972948
&5042951797248
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 2 $}

\subsection{Partition $1$ of model $2$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&1&1&0&0&1&0
\\\noalign{\medskip}0&1&1&1&0&0&1&1&0&2\\\noalign{\medskip}0&0&0&0&1&0
&0&0&-1&0\\\noalign{\medskip}0&0&0&-2&0&1&0&0&0&-1\\\noalign{\medskip}0
&0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&1&0&0&0&1&0
\\\noalign{\medskip}-1&-2&0&0&0&1&0&0&0&1\\\noalign{\medskip}0&0&1&1&0
&0&1&1&0&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{4}}
$, $K_{{1}}
=D_{{3}}
$, $K_{{3}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{4}}D_{{8}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{2}}}^{2}+4\,K_{{2}}K_{{1}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 58 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&1&0&0&0&1&0
\\\noalign{\medskip}-2&-4&1&1&0&2&1&1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)={J_{{1}}}^{2}+6\,{J_{{2}}}^{2}+4\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=40
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&768
&9376
&293632
&10924128
&495966976
\\ 
1 &16
&2176
&234176
&24185856
&2411703648
&234707597824
\\ 
2 &0
&768
&495360
&168619008
&40145749504
&7766302256640
\\ 
3 &0
&0
&234176
&297281152
&172102715744
&66068613443584
\\ 
4 &0
&0
&9376
&168619008
&270657325696
&214069150771200
\\ 
5 &0
&0
&0
&24185856
&172102715744
&311660717547904
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 14 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&0&0&1&0&0&1&1&1&1&1
\\\noalign{\medskip}0&1&1&1&0&1&1&0&0&0&1&1\\\noalign{\medskip}0&0&0&1
&1&0&0&0&1&-1&0&1\\\noalign{\medskip}0&0&0&-2&0&0&1&0&0&0&-1&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0&0&
-1&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&-1&-1&0&0&0&-1&0&0&1&1
\\\noalign{\medskip}0&-1&0&0&-1&1&0&0&1&0&0&0\\\noalign{\medskip}0&-2&0
&0&-1&0&1&0&0&0&0&1\\\noalign{\medskip}0&0&1&1&1&0&0&1&0&0&0&-2
\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&1&-2&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{5}}
=D_{{8}}
$, $K_{{3}}
=D_{{5}}
$, $K_{{2}}
=D_{{4}}
$, $K_{{4}}
=D_{{3}}+D_{{4}}+D_{{5}}
$, $K_{{1}}
=D_{{3}}+D_{{4}}+D_{{5}}-D_{{1}}+D_{{8}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{4}}D_{{7}},D_{{5}}D_{{9}},D_{{5}}D_
{{10}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_
{{1}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{6}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{2}}K_{{1}}+2\,K_{{3}}K_{{1}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 58 $.
The divisors $D_{{3}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&-1&0&0&0&1&0&0&1&1&0&0
\\\noalign{\medskip}-2&-4&1&1&0&0&2&1&0&1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{2}}
\qquad
J_{{2}}=1/2\,K_{{1}}-K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{2}}}^{2}+7\,J_{{2}}J_{{1}}+{J_{{1}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=1
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=34
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&768
&9376
&293632
&10924128
&495966976
\\ 
1 &64
&7040
&748608
&76702592
&7591063808
&735153731328
\\ 
2 &24
&24960
&10131280
&2746039040
&576628466328
&103124584231552
\\ 
3 &0
&51072
&66239296
&40452980224
&16240157919808
&4984227552464768
\\ 
4 &0
&63520
&267646080
&344221451008
&247778295207936
&124048877725156992
\\ 
5 &0
&51072
&741806144
&1962255923584
&2443092985545600
&1928383581765485696
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $ in the list}

\subsection{Partition $1$ of model $1$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&1&1&1&0
\\\noalign{\medskip}0&1&1&1&1&0&0&0&1\\\noalign{\medskip}0&0&1&0&0&-1&0
&0&1\\\noalign{\medskip}0&0&1&0&1&-1&0&0&0\\\noalign{\medskip}0&0&-1&0
&0&0&1&0&0\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-2&0&0&1&1&0&0&1
\\\noalign{\medskip}-2&0&1&1&-1&0&1&1&-1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{1}}
$, $K_{{1}}
=D_{{1}}+D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{4}}D_{{7}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{1}}}^{2}+4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 60 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-2&0&0&1&1&0&0&1
\\\noalign{\medskip}-3&-2&1&1&0&1&1&1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}-K_{{2}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)={J_{{1}}}^{2}+6\,J_{{2}}J_{{1}}+4\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=6
&
K_{{2,2,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=44
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&384
&2634
&35584
&606012
&12211584
\\ 
1 &72
&2736
&130392
&5764032
&249232392
&10571395104
\\ 
2 &18
&6912
&1243224
&144080640
&12971756646
&999171770112
\\ 
3 &0
&9440
&5603256
&1480459440
&254979255792
&33534668602080
\\ 
4 &0
&6912
&15055758
&8623900800
&2694436483392
&580400203457664
\\ 
5 &0
&2736
&26470224
&32830966656
&18124556932656
&6221352556523088
\\ 
\hline
\end{array}
\end{equation}


\section{Polyhedron $ 4 $}

\subsection{Partition $1$ of model $4$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&1&0&0&1&0&1
\\\noalign{\medskip}0&1&1&0&1&1&0&1&0\\\noalign{\medskip}0&0&1&0&0&0&0
&1&-1\\\noalign{\medskip}0&0&0&0&1&0&0&0&-1\\\noalign{\medskip}0&0&0&0
&0&1&0&0&-1\\\noalign{\medskip}0&0&0&0&0&0&1&0&-1\\\noalign{\medskip}0
&0&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&0&1&1&0&0&0&-1&0
\\\noalign{\medskip}-2&-3&0&0&1&1&1&1&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{1}}
$, $K_{{2}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{3}}D_{{4}}D_{{5}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
3\,K_{{1}}K_{{2}}+6\,{K_{{2}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 62 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&0&1&1&0&0&0&-1&0
\\\noalign{\medskip}-2&-3&0&0&1&1&1&1&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=-{J_{{1}}}^{2}+4\,{J_{{2}}}^{2}+3\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=6
&
K_{{2,2,2}}
&=9
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=54
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&378
&2646
&35556
&606096
&12211290
\\ 
1 &12
&648
&30132
&1353696
&59046084
&2521375776
\\ 
2 &0
&27
&19764
&3622821
&405431784
&35520639453
\\ 
3 &0
&0
&324
&1398216
&529944336
&106249923936
\\ 
4 &0
&0
&-54
&15762
&143215128
&87235305480
\\ 
5 &0
&0
&0
&-1968
&1316952
&18108696960
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&0&1&1&1
\\\noalign{\medskip}0&1&1&1&1&1&0&0&0\\\noalign{\medskip}0&0&0&0&1&0&0
&0&-1\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&1&1&0&1&1&1&0
\\\noalign{\medskip}-1&-1&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
6\,{K_{{1}}}^{2}+5\,K_{{1}}K_{{2}}+{K_{{2}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 66 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&1&1&0&1&1&1&0
\\\noalign{\medskip}-1&-1&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=4\,{J_{{1}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=5
&
K_{{1,1,2}}
&=6
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=50
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&36
&0
&0
&0
&0
\\ 
1 &366
&1584
&909
&16
&0
&0
\\ 
2 &2670
&73728
&255960
&231336
&45216
&360
\\ 
3 &35500
&3286224
&34736049
&106245024
&119474748
&48046176
\\ 
4 &606264
&142523712
&3387935304
&23702767680
&66922830504
&85607985132
\\ 
5 &12210702
&6060689280
&273906849222
&3623779411776
&19938817169442
&53346064121712
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

\subsection{Partition $1$ of model $1$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&1&1&1&0
\\\noalign{\medskip}0&1&1&1&1&0&0&0&1\\\noalign{\medskip}0&0&0&-2&0&0&0
&1&-1\\\noalign{\medskip}0&0&0&-2&1&0&0&0&-1\\\noalign{\medskip}0&0&0&
-2&0&1&0&0&-1\\\noalign{\medskip}0&0&0&-2&0&0&1&0&-1
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -3&-2&0&0&1&1&1&1&1
\\\noalign{\medskip}0&0&1&1&0&0&0&0&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{3}}
$, $K_{{2}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{3}}D_{{4}}D_{{5}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
6\,{K_{{1}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 68 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -3&-2&0&0&1&1&1&1&1
\\\noalign{\medskip}0&0&1&1&0&0&0&0&-2\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=-3\,{J_{{2}}}^{2}+4\,{J_{{1}}}^{2}+2\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=12
&
K_{{1,1,2}}
&=6
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=60
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&6
&0
&0
&0
&0
\\ 
1 &360
&360
&0
&0
&0
&0
\\ 
2 &2682
&17064
&2682
&0
&0
&0
\\ 
3 &35472
&770280
&770280
&35472
&0
&0
\\ 
4 &606348
&33726420
&99533664
&33726420
&606348
&0
\\ 
5 &12210408
&1444231296
&9382024152
&9382024152
&1444231296
&12210408
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

\subsection{Partition $1$ of model $1$}
The model has $5$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&1&0&0&1&0&1&1
\\\noalign{\medskip}0&1&1&0&1&1&0&1&0&1\\\noalign{\medskip}0&0&-1&0&0&0
&0&1&-2&-1\\\noalign{\medskip}0&0&-1&0&1&0&0&0&-1&0
\\\noalign{\medskip}0&0&-1&0&0&1&0&0&-1&0\\\noalign{\medskip}0&0&-1&0&0
&0&1&0&-1&0\\\noalign{\medskip}0&0&-1&1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -1&-2&1&1&0&0&0&0&-1&1
\\\noalign{\medskip}0&0&-1&-1&0&0&0&1&1&0\\\noalign{\medskip}0&0&0&0&1
&1&1&0&1&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{1}}+D_{{6}}
$, $K_{{2}}
=D_{{6}}
$, $K_{{3}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{6}}D_{{7}},D_{{6}}D_{{8}},D_{{1}}D_{{2}}D_{{8}},D_{{3}}D_{{4}}D_{
{5}}D_{{7}},D_{{1}}D_{{2}}D_{{3}}D_{{4}}D_{{5}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{1}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-4&2&2&1&1&1&0&-1&0
\\\noalign{\medskip}0&0&-1&-1&0&0&0&1&1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=4\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=6
&
K_{{1,1,2}}
&=8
&
K_{{1,2,2}}
&=8
&
K_{{2,2,2}}
&=8
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=48
&
\ch_2 \cdot J_2 &=56
&
\end{align}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&8
&0
&0
&0
&0
\\ 
1 &56
&1184
&704
&0
&0
&0
\\ 
2 &-272
&1104
&82832
&130192
&9704
&0
\\ 
3 &3240
&-18176
&132544
&13865248
&36973952
&13506752
\\ 
4 &-58432
&400536
&-2699264
&26457848
&3385637232
&12854281944
\\ 
5 &1303840
&-10532352
&71413056
&-577466880
&7219059360
&1021710328800
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&1&0&0&1&0&1&1
\\\noalign{\medskip}0&1&1&0&1&1&0&1&0&1\\\noalign{\medskip}0&0&-1&0&0&0
&0&1&-2&-1\\\noalign{\medskip}0&0&-1&0&1&0&0&0&-1&0
\\\noalign{\medskip}0&0&-1&0&0&1&0&0&-1&0\\\noalign{\medskip}0&0&-1&0&0
&0&1&0&-1&0\\\noalign{\medskip}0&0&-1&1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -1&-2&0&0&0&0&0&1&0&1
\\\noalign{\medskip}0&0&1&1&0&0&0&-1&-1&0\\\noalign{\medskip}0&0&0&0&1
&1&1&0&1&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{1}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{1}}
=D_{{6}}+D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{6}}D_{{8}},D_{{3}}D_{{4}}D_{{5}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{1}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-4&0&0&1&1&1&2&1&0
\\\noalign{\medskip}0&0&1&1&0&0&0&-1&-1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+3\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=6
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=48
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&8
&0
&0
&0
&0
\\ 
1 &704
&1184
&56
&0
&0
&0
\\ 
2 &9704
&130192
&82832
&1104
&-272
&0
\\ 
3 &291008
&13506752
&36973952
&13865248
&132544
&-18176
\\ 
4 &10952056
&1351951736
&9855970048
&12854281944
&3385637232
&26457848
\\ 
5 &495618240
&131926687008
&2028056415104
&6201762710528
&5040816284800
&1021710328800
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 2 $}

\subsection{Partition $1$ of model $2$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&1&0&0&0&1&1
\\\noalign{\medskip}0&1&1&0&1&1&1&0&0\\\noalign{\medskip}0&0&1&0&0&0&0
&1&-1\\\noalign{\medskip}0&0&0&0&1&0&0&0&-1\\\noalign{\medskip}0&0&0&0
&0&1&0&0&-1\\\noalign{\medskip}0&0&0&0&0&0&1&0&-1\\\noalign{\medskip}0
&0&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&0&0&1&1&1&1&1
\\\noalign{\medskip}0&-1&1&1&0&0&0&-1&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{3}}
$, $K_{{2}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{3}}D_{{4}}D_{{5}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
6\,{K_{{1}}}^{2}+2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&0&0&1&1&1&1&1
\\\noalign{\medskip}0&-1&1&1&0&0&0&-1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=-{J_{{2}}}^{2}+4\,{J_{{1}}}^{2}+4\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=8
&
K_{{1,1,2}}
&=6
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=56
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&18
&0
&0
&0
&0
\\ 
1 &348
&900
&36
&-4
&0
&0
\\ 
2 &2706
&41778
&46548
&1512
&-306
&54
\\ 
3 &35416
&1871784
&8009712
&5604204
&153936
&-24768
\\ 
4 &606516
&81468792
&864795636
&1928672640
&985016556
&25990110
\\ 
5 &12209820
&3473471196
&74041264872
&363480492960
&530436671676
&214272257040
\\ 
\hline
\end{array}
\end{equation}


\section{Polyhedron $ 4 $}

\subsection{Partition $1$ of model $4$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&0&1&0&0&1&0
\\\noalign{\medskip}0&1&1&1&1&0&1&1&0&2\\\noalign{\medskip}0&0&0&0&1&0
&0&0&-1&0\\\noalign{\medskip}0&0&0&-2&0&1&0&0&0&-1\\\noalign{\medskip}0
&0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -1&-1&0&0&1&0&0&0&1&0
\\\noalign{\medskip}-1&-2&0&0&0&1&0&0&0&1\\\noalign{\medskip}0&0&1&1&0
&0&1&1&0&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{3}}
$, $K_{{3}}
=D_{{1}}
$, $K_{{2}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{4}}D_{{8}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
{K_{{1}}}^{2}+3\,K_{{1}}K_{{2}}+2\,{K_{{2}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -1&-1&0&0&1&0&0&0&1&0
\\\noalign{\medskip}-2&-4&1&1&0&2&1&1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{2}}}^{2}+6\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=3
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=42
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&704
&9704
&291008
&10952056
&495618240
\\ 
1 &32
&3904
&416736
&42873024
&4258889312
&413449786176
\\ 
2 &0
&3216
&2030176
&644810720
&146713202496
&27524945558992
\\ 
3 &0
&64
&2699616
&2830008448
&1466448925952
&522305452066176
\\ 
4 &0
&-4
&860352
&4743969216
&6016375594432
&4136767940274624
\\ 
5 &0
&0
&12768
&3034501632
&11540204027808
&16213245681158080
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 16 $}

\subsection{Partition $1$ of model $16$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&1&0&0&0&0&1&0&1&0&0
\\\noalign{\medskip}0&1&0&1&1&1&1&0&1&0&2&2\\\noalign{\medskip}0&0&0&3
&-1&0&0&0&1&0&1&2\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&0&0&0
\\\noalign{\medskip}0&0&0&-1&0&1&0&0&0&0&0&0\\\noalign{\medskip}0&0&1&
-2&0&0&0&1&0&1&-1&-1\\\noalign{\medskip}0&0&2&-2&0&0&0&0&0&1&-1&-1
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&0&0&0&0&0&0&-2&1&0&1
\\\noalign{\medskip}0&-1&0&-1&0&-1&-1&0&0&0&1&1\\\noalign{\medskip}0&0
&1&0&0&0&0&1&0&-2&0&0\\\noalign{\medskip}0&0&0&1&0&1&1&0&1&0&0&-2
\\\noalign{\medskip}0&0&0&1&1&1&1&0&0&0&-2&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{5}}
=D_{{3}}
$, $K_{{3}}
=D_{{1}}
$, $K_{{1}}
=D_{{8}}+2\,D_{{1}}
$, $K_{{4}}
=D_{{7}}+2\,D_{{8}}+4\,D_{{1}}
$, $K_{{2}}
=-D_{{2}}+D_{{3}}+D_{{7}}+2\,D_{{8}}+4\,D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{6}},D_{{3}}D_{{7}},D_{{3}}D_{{10}},D_{{7}}D_{{9}},D_{{8}}D
_{{9}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{2}}D_{{3}}D_{{4}}D_{{5}},D_
{{2}}D_{{4}}D_{{5}}D_{{7}},D_{{2}}D_{{4}}D_{{5}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{7}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&-6&0&1&3&1&1&0&0&2&0&0
\\\noalign{\medskip}0&0&1&0&0&0&0&1&0&-2&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{3}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{1}}}^{2}+4\,J_{{2}}J_{{1}}-3\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=2
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=32
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&8
&0
&0
&0
&0
\\ 
1 &3264
&9024
&3264
&0
&0
&0
\\ 
2 &193704
&6677856
&14161056
&6677856
&193704
&0
\\ 
3 &40897216
&4501790208
&33088733952
&58621711936
&33088733952
&4501790208
\\ 
4 &8869331832
&2899431143088
&53279323863936
&234194824987824
&369911930813472
&234194824987824
\\ 
5 &2585556116160
&1809306027051264
&68666260414628736
&614033520793122816
&2022411513672681984
&2955264922570038720
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 19 $ in the list}

\subsection{Partition $1$ of model $19$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&1&0&1&0&1&1
\\\noalign{\medskip}0&1&1&1&0&1&0&1&0&0\\\noalign{\medskip}0&0&0&-2&1&0
&0&-1&0&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&-1&1&0
\\\noalign{\medskip}0&0&0&0&0&1&0&-1&0&0\\\noalign{\medskip}0&0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&1&1&1&-1&1&-1&0&0
\\\noalign{\medskip}0&-2&1&1&0&0&1&0&1&-2\\\noalign{\medskip}0&0&-1&-1
&0&1&-1&1&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{3}}
$, $K_{{3}}
=D_{{4}}+D_{{3}}
$, $K_{{2}}
=D_{{1}}+D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{4}}D_{{6}}D_{{7}},D_{{4}}D_{{6}}D_{{8}},D_{{1}}D_{
{2}}D_{{3}}D_{{5}},D_{{1}}D_{{2}}D_{{5}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -4&-2&1&1&2&0&1&0&1&0
\\\noalign{\medskip}-4&-4&0&0&2&2&0&2&2&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=2\,K_{{1}}-K_{{2}}
\qquad
J_{{2}}=-K_{{1}}+K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=3\,J_{{2}}J_{{1}}+2\,{J_{{1}}}^{2}-3\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=108
&
K_{{1,1,2}}
&=-24
&
K_{{1,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=132
&
\ch_2 \cdot J_2 &=-36
&
\end{align}
$J_{{2}}
$ is an elliptic surface.
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&1&0&1&0&1&1
\\\noalign{\medskip}0&1&1&1&0&1&0&1&0&0\\\noalign{\medskip}0&0&0&-2&1&0
&0&-1&0&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&-1&1&0
\\\noalign{\medskip}0&0&0&0&0&1&0&-1&0&0\\\noalign{\medskip}0&0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&1&0&0&0&0&1
\\\noalign{\medskip}0&-2&0&0&0&1&0&1&1&-1\\\noalign{\medskip}0&0&1&1&0
&-1&1&-1&0&-1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{3}}
=D_{{1}}
$, $K_{{2}}
=D_{{4}}+D_{{1}}
$, $K_{{1}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{1}}D_{{2}}D_{{5}},D_{{4}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-2&0&0&1&1&0&1&1&0
\\\noalign{\medskip}-4&-2&1&1&2&0&1&0&1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=2\,K_{{2}}-K_{{1}}
\qquad
J_{{2}}=K_{{1}}-K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=9\,J_{{2}}J_{{1}}+2\,{J_{{1}}}^{2}+6\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=36
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
$J_{{2}}
$ is an elliptic surface.
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&704
&9704
&291008
&10952056
&495618240
\\ 
1 &128
&12288
&1337344
&135708672
&13397379072
&1294300782592
\\ 
2 &144
&103936
&39945216
&10230360064
&2071519497472
&361095490311168
\\ 
3 &128
&638976
&658513920
&353338880000
&130539974270976
&37773988952834048
\\ 
4 &140
&3173888
&7635363648
&7724662999040
&4810652504762560
&2177625853806633984
\\ 
5 &128
&13639680
&69465993216
&123660569706496
&122880050549305344
&83266978978116784128
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 28 $ in the list}

\subsection{Partition $1$ of model $28$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} 1&0&0&1&0&1&0&0&1&0&1&1&1
\\\noalign{\medskip}0&1&1&0&1&0&1&1&0&1&1&1&1\\\noalign{\medskip}0&0&0
&-3&1&0&0&0&-1&-1&-2&-2&-1\\\noalign{\medskip}0&0&0&-2&0&0&0&1&0&0&-1&
-1&-1\\\noalign{\medskip}0&0&0&-1&0&1&0&0&-1&0&-1&0&0
\\\noalign{\medskip}0&0&0&0&0&0&1&0&-1&0&0&0&0\\\noalign{\medskip}0&0&
1&-1&0&0&0&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -1&0&0&0&0&0&0&1&0&-2&0&1&0
\\\noalign{\medskip}0&-1&-1&-1&0&-1&0&0&0&0&0&1&1\\\noalign{\medskip}0
&-2&0&0&1&0&-1&1&-1&0&1&0&0\\\noalign{\medskip}0&0&1&1&0&0&1&0&1&0&-2&0
&0\\\noalign{\medskip}0&0&1&1&1&1&0&0&0&0&0&0&-2\\\noalign{\medskip}0&0
&1&1&0&1&0&0&0&1&0&-2&0\\\noalign{\medskip}0&0&-1&-1&0&0&0&-1&0&1&1&0&0
\end {array} \right] 
\end{equation}
This Mori cone is not simplicial. 
A positive basis for the generators of the K\"ahler cone is $K_{{6}}
=1/2\,D_{{8}}+D_{{6}}-3/2\,D_{{1}}+3/2\,D_{{5}}+D_{{4}}+1/2\,D_{{3}}
$, $K_{{5}}
=-1/2\,D_{{3}}-1/2\,D_{{1}}+1/2\,D_{{5}}+1/2\,D_{{8}}+D_{{6}}+D_{{4}}
$, $K_{{2}}
=-1/2\,D_{{1}}+1/2\,D_{{5}}+1/2\,D_{{8}}+D_{{6}}+1/2\,D_{{3}}
$, $K_{{1}}
=D_{{6}}-D_{{1}}+D_{{5}}+D_{{4}}
$, $K_{{4}}
=1/2\,D_{{3}}+1/2\,D_{{1}}-1/2\,D_{{5}}-1/2\,D_{{8}}-D_{{6}}
$, $K_{{3}}
=3/2\,D_{{5}}-1/2\,D_{{1}}+1/2\,D_{{8}}+D_{{6}}+1/2\,D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{10}},D_{{4}}D_{{9}},D_{{6}}D_{{10}},D_{{6}}
D_{{11}},D_{{8}}D_{{9}},D_{{8}}D_{{11}},D_{{9}}D_{{10}},D_{{9}}D_{{11}
},D_{{10}}D_{{11}},D_{{3}}D_{{6}}D_{{9}},D_{{4}}D_{{5}}D_{{7}},D_{{5}}
D_{{7}}D_{{8}},D_{{5}}D_{{7}}D_{{10}},D_{{5}}D_{{7}}D_{{11}},D_{{1}}D_
{{2}}D_{{3}}D_{{4}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_{{1}}D_{{2}}D_{{4}}
D_{{6}},D_{{1}}D_{{2}}D_{{4}}D_{{8}},D_{{1}}D_{{2}}D_{{5}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -4&-12&2&2&6&2&0&4&0&0&0&0&0
\\\noalign{\medskip}-12&-12&0&0&6&6&6&0&6&0&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{2}}-1/4\,K_{{1}}
\qquad
J_{{2}}=-1/2\,K_{{2}}+1/4\,K_{{1}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=16\,J_{{2}}J_{{1}}+8\,{J_{{1}}}^{2}+2\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&={\frac {75}{4}}
&
K_{{1,1,2}}
&=-5
&
K_{{1,2,2}}
&=1
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=72
&
\ch_2 \cdot J_2 &=-24
&
\end{align}
$J_{{2}}
$ is an elliptic surface.
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} 1&0&0&1&0&1&0&0&1&0&1&1&1
\\\noalign{\medskip}0&1&1&0&1&0&1&1&0&1&1&1&1\\\noalign{\medskip}0&0&0
&-3&1&0&0&0&-1&-1&-2&-2&-1\\\noalign{\medskip}0&0&0&-2&0&0&0&1&0&0&-1&
-1&-1\\\noalign{\medskip}0&0&0&-1&0&1&0&0&-1&0&-1&0&0
\\\noalign{\medskip}0&0&0&0&0&0&1&0&-1&0&0&0&0\\\noalign{\medskip}0&0&
1&-1&0&0&0&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -1&0&-1&-1&0&0&0&0&0&-1&1&1&0
\\\noalign{\medskip}0&-1&-1&-1&0&-1&0&0&0&0&0&1&1\\\noalign{\medskip}0
&0&1&1&1&1&0&0&0&0&0&0&-2\\\noalign{\medskip}0&0&1&1&0&1&0&0&0&1&0&-2&0
\\\noalign{\medskip}0&0&1&1&0&0&1&0&1&0&-2&0&0\\\noalign{\medskip}0&0&0
&0&0&0&-1&1&-1&-1&1&0&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{6}}
=D_{{6}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{5}}
=D_{{5}}+D_{{6}}
$, $K_{{4}}
=D_{{4}}+D_{{8}}-D_{{1}}+D_{{5}}+2\,D_{{6}}
$, $K_{{2}}
=D_{{8}}-D_{{1}}+D_{{5}}+2\,D_{{6}}+D_{{3}}
$, $K_{{1}}
=-D_{{1}}+D_{{4}}+D_{{5}}+D_{{6}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{10}},D_{{4}}D_{{9}},D_{{6}}D_{{9}},D_{{6}}D
_{{10}},D_{{6}}D_{{11}},D_{{8}}D_{{11}},D_{{9}}D_{{10}},D_{{9}}D_{{11}
},D_{{10}}D_{{11}},D_{{1}}D_{{2}}D_{{6}},D_{{4}}D_{{5}}D_{{7}},D_{{5}}
D_{{7}}D_{{10}},D_{{5}}D_{{7}}D_{{11}},D_{{1}}D_{{2}}D_{{3}}D_{{4}},D_
{{1}}D_{{2}}D_{{4}}D_{{8}},D_{{1}}D_{{2}}D_{{5}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 72 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -2&-2&0&0&1&1&1&0&1&0&0&0&0
\\\noalign{\medskip}-2&-6&1&1&3&1&0&2&0&0&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=3/4\,K_{{1}}-1/4\,K_{{2}}
\qquad
J_{{2}}=-1/4\,K_{{1}}+1/4\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,J_{{2}}J_{{1}}+2\,{J_{{1}}}^{2}+12\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 0\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&3264
&193704
&40897216
&8869331832
&2585556116160
\\ 
1 &96
&84480
&58787328
&39090493440
&24853336094208
&15384774694677504
\\ 
2 &148
&1048064
&2731077824
&4777961654272
&6504047417702720
&7526789771866457088
\\ 
3 &96
&9029632
&66941515776
&252439380582912
&643410594137323520
&1267240664887775686656
\\ 
4 &140
&60521984
&1121769568832
&8212311229630464
&36029225481582655168
&113097188395002902791168
\\ 
5 &96
&341040128
&14408722875392
&191754703069319168
&1370287344975752832000
&6545198493349608639575552
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&0&1&1&0
\\\noalign{\medskip}0&1&1&1&1&1&0&0&1\\\noalign{\medskip}0&0&0&0&1&0&0
&0&-1\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&1&1&0&1&1&1&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
6\,{K_{{1}}}^{2}+4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 76 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -2&-3&1&1&0&1&1&1&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)={J_{{2}}}^{2}+4\,{J_{{1}}}^{2}+6\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=4
&
K_{{1,1,2}}
&=6
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=52
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 6\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&54
&0
&0
&0
&0
\\ 
1 &336
&2160
&2160
&336
&0
&0
\\ 
2 &2730
&102492
&583470
&1011384
&583470
&102492
\\ 
3 &35360
&4539024
&75358080
&400151088
&885946464
&885946464
\\ 
4 &606684
&196580898
&7141800672
&82704010218
&418964809644
&1066633771068
\\ 
5 &12209232
&8346574800
&566341955568
&12034210855968
&113289157975680
&555840165325248
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 1 $}

\subsection{Partition $1$ of model $1$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&1&0&0&1&1
\\\noalign{\medskip}0&1&1&1&0&1&1&0&0\\\noalign{\medskip}0&0&0&0&1&0&0
&0&-1\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-4&1&1&0&1&1&1&0
\\\noalign{\medskip}-2&0&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,{K_{{1}}}^{2}+8\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 86 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-4&1&1&0&1&1&1&0
\\\noalign{\medskip}-2&0&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+{J_{{2}}}^{2}+2\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=8
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=56
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&4
&0
&0
&0
&0
\\ 
1 &640
&640
&0
&0
&0
&0
\\ 
2 &10032
&72224
&10032
&0
&0
&0
\\ 
3 &288384
&7539200
&7539200
&288384
&0
&0
\\ 
4 &10979984
&757561520
&2346819520
&757561520
&10979984
&0
\\ 
5 &495269504
&74132328704
&520834042880
&520834042880
&74132328704
&495269504
\\ 
\hline
\end{array}
\end{equation}
\subsection{Partition $2$ of model $1$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccc} 1&0&0&0&0&0&0&1&1
\\\noalign{\medskip}0&1&1&1&1&1&1&0&0\\\noalign{\medskip}0&0&0&0&1&0&0
&0&-1\\\noalign{\medskip}0&0&-1&0&0&0&0&1&0\\\noalign{\medskip}0&0&-1&0
&0&1&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0
&-1&1&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-4&1&1&0&1&1&1&0
\\\noalign{\medskip}-1&-1&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,{K_{{1}}}^{2}+5\,K_{{1}}K_{{2}}+{K_{{2}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 86 $.
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccc} -1&-4&1&1&0&1&1&1&0
\\\noalign{\medskip}-1&-1&0&0&1&0&0&0&1\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=5
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=50
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&16
&0
&0
&0
&0
\\ 
1 &640
&2144
&120
&-32
&3
&0
\\ 
2 &10032
&231888
&356368
&14608
&-4920
&1680
\\ 
3 &288384
&23953120
&144785584
&144051072
&5273880
&-1505472
\\ 
4 &10979984
&2388434784
&36512550816
&115675981232
&85456640608
&3018009984
\\ 
5 &495269504
&232460466048
&7251261673320
&50833652046112
&106397389165188
&62800738246496
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 6 $}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&0&0&1&1&0&1
\\\noalign{\medskip}0&1&1&1&1&1&0&0&1&1\\\noalign{\medskip}0&0&0&0&1&0
&0&0&-1&0\\\noalign{\medskip}0&0&0&-2&0&1&0&0&0&-1\\\noalign{\medskip}0
&0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -1&-2&0&0&0&1&0&0&0&1
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1&0\\\noalign{\medskip}0&0&1&1&0&0
&1&1&0&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{3}}
=D_{{1}}
$, $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{7}},D_{{4}}D_{{8}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{1}}}^{2}+2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 86 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-4&1&1&0&2&1&1&0&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+{J_{{2}}}^{2}+8\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=2
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=44
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&64
&0
&0
&0
&0
\\ 
1 &640
&6912
&14400
&6912
&640
&0
\\ 
2 &10032
&742784
&8271360
&31344000
&48098560
&31344000
\\ 
3 &288384
&75933184
&2445747712
&26556152064
&130867460608
&329212616704
\\ 
4 &10979984
&7518494784
&532817161216
&12305418469184
&130700405114112
&746592735013952
\\ 
5 &495269504
&728114777344
&97089446866176
&4074651399444224
&78142574531195136
&816759204484794624
\\ 
\hline
\end{array}
\end{equation}



\section{Polyhedron $ 4 $}

\subsection{Partition $1$ of model $4$}
The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&1&0&0&0&1&1&0
\\\noalign{\medskip}0&1&1&0&1&1&1&0&0&1\\\noalign{\medskip}0&0&0&-3&1&0
&0&-2&0&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&-1&1&0
\\\noalign{\medskip}0&0&0&-1&0&1&0&-1&0&0\\\noalign{\medskip}0&0&0&-1&0
&0&1&-1&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&1&1&0&1&1&0&1&-3
\\\noalign{\medskip}0&-2&1&1&1&0&0&-1&0&0\\\noalign{\medskip}0&0&-1&-1
&0&0&0&1&0&1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{3}}
=-D_{{1}}+D_{{4}}+D_{{3}}
$, $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{6}}D_{{8}},D_{{1}}D_{{2}}D_{{3}},D_{{4}}D_{{5}}D_{
{6}}D_{{7}},D_{{1}}D_{{2}}D_{{4}}D_{{5}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-4&0&0&2&1&1&1&1&0
\\\noalign{\medskip}0&-2&1&1&1&0&0&-1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=-2\,K_{{1}}+K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+7\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=4
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=52
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&32
&-2
&0
&0
&0
\\ 
1 &576
&3840
&576
&0
&0
&0
\\ 
2 &10360
&413056
&1533744
&445824
&-55176
&98304
\\ 
3 &285760
&42452992
&561744128
&1518375680
&580618496
&4704256
\\ 
4 &11007912
&4217952960
&134320808616
&1028366160576
&2261335597152
&1046296810176
\\ 
5 &494920768
&409491950592
&25791106777728
&411041211686912
&2228121584016896
&4243661500095744
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&1&0&0&0&1&1&0
\\\noalign{\medskip}0&1&1&0&1&1&1&0&0&1\\\noalign{\medskip}0&0&0&-3&1&0
&0&-2&0&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&-1&1&0
\\\noalign{\medskip}0&0&0&-1&0&1&0&-1&0&0\\\noalign{\medskip}0&0&0&-1&0
&0&1&-1&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&0&1&1&1&1&-2
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&0&1\\\noalign{\medskip}0&0&1&1&0&0
&0&-1&0&-1\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{1}}
=D_{{4}}
$, $K_{{3}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{3}}D_{{8}},D_{{4}}D_{{5}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
4\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-4&0&0&2&1&1&1&1&0
\\\noalign{\medskip}0&-2&1&1&1&0&0&-1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=-2\,K_{{1}}+K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=6\,{J_{{1}}}^{2}+7\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=4
&
K_{{1,1,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=52
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 4\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&32
&-2
&0
&0
&0
\\ 
1 &576
&3840
&576
&0
&0
&0
\\ 
2 &10360
&413056
&1533744
&445824
&-55176
&98304
\\ 
3 &285760
&42452992
&561744128
&1518375680
&580618496
&4704256
\\ 
4 &11007912
&4217952960
&134320808616
&1028366160576
&2261335597152
&1046296810176
\\ 
5 &494920768
&409491950592
&25791106777728
&411041211686912
&2228121584016896
&4243661500095744
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 5 $}

\subsection{Partition $1$ of model $5$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&1&0&1&0&0&0&0&0&0&0
\\\noalign{\medskip}0&1&0&1&0&1&1&1&1&1&2&2\\\noalign{\medskip}0&0&0&0
&0&-1&0&0&3&1&1&2\\\noalign{\medskip}0&0&0&0&1&0&0&0&-2&0&-1&-1
\\\noalign{\medskip}0&0&1&0&0&0&0&1&-1&0&0&0\\\noalign{\medskip}0&0&0&0
&0&0&1&0&-1&0&0&0\\\noalign{\medskip}0&0&-1&1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&1&1&0&0&0&-1&0&0&0&0
\\\noalign{\medskip}-1&0&0&0&1&0&0&0&0&-2&0&1\\\noalign{\medskip}0&-1&0
&0&0&0&-1&-1&-1&0&1&1\\\noalign{\medskip}0&0&0&0&0&1&1&1&1&0&-2&0
\\\noalign{\medskip}0&0&0&0&0&0&1&1&1&1&0&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{3}}
$, $K_{{4}}
=D_{{4}}
$, $K_{{3}}
=-D_{{5}}+D_{{4}}+D_{{8}}+2\,D_{{3}}
$, $K_{{5}}
=D_{{8}}+2\,D_{{3}}
$, $K_{{1}}
=D_{{1}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{3}}D_{{9}},D_{{3}}D_{{10}},D_{{4}}D_{{8}},D_{{4}}D
_{{10}},D_{{8}}D_{{9}},D_{{9}}D_{{10}},D_{{3}}D_{{5}}D_{{6}}D_{{7}},D_
{{4}}D_{{5}}D_{{6}}D_{{7}},D_{{5}}D_{{6}}D_{{7}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{3}}+K_{{2}}K_{{3}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&1&1&0&0&0&-1&0&0&0&0
\\\noalign{\medskip}-2&-6&0&0&2&3&1&1&1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{2}}}^{2}-{J_{{1}}}^{2}+3\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=2
&
K_{{2,2,2}}
&=3
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=42
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&2880
&208728
&39767360
&8984778408
&2571518571840
\\ 
1 &4
&4752
&3603528
&2445085664
&1580414726844
&988726084819920
\\ 
2 &0
&252
&2204928
&6905771456
&12536489815488
&17213021137437120
\\ 
3 &0
&0
&20520
&2493967312
&16303941022572
&54281784750359040
\\ 
4 &0
&0
&-9252
&17671840
&4149078701904
&43526767714418880
\\ 
5 &0
&0
&0
&-2695040
&24150423420
&8540389793334960
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 8 $}

\subsection{Partition $1$ of model $8$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&1&1&0&0&0&0&0&0&1&1
\\\noalign{\medskip}0&1&0&0&1&1&1&1&1&1&1&1\\\noalign{\medskip}0&0&0&-
3&1&0&0&0&-1&-1&-2&-1\\\noalign{\medskip}0&0&0&-2&0&0&1&0&1&0&-1&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0&0&
-1&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&0&0&0&1&0&0&1&-3&1&0
\\\noalign{\medskip}0&-2&0&0&1&1&-1&0&1&0&0&0\\\noalign{\medskip}0&-1&
-1&-1&0&0&0&-1&0&0&1&1\\\noalign{\medskip}0&0&1&1&1&0&0&1&0&0&0&-2
\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&1&-2&0\\\noalign{\medskip}0&0&0&0
&0&-1&1&0&-1&1&0&0\end {array} \right] 
\end{equation}
This Mori cone is not simplicial. 
A positive basis for the generators of the K\"ahler cone is $K_{{5}}
=D_{{5}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{1}}
=D_{{4}}+D_{{5}}
$, $K_{{4}}
=D_{{8}}+3\,D_{{4}}+2\,D_{{5}}
$, $K_{{2}}
=-D_{{1}}+2\,D_{{5}}+D_{{3}}+D_{{8}}+3\,D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{5}}D_{{8}},D_{{5}}D_{{9}},D_{{5}}D_
{{10}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{3}}D_{{4}}D_{{7}},D_{{4}}D_
{{7}}D_{{9}},D_{{4}}D_{{7}}D_{{10}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_{{1
}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{6}}D_{{8}},D_{{1}}D_{{2}}D_{
{4}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&-6&1&1&3&0&2&1&0&0&0&0
\\\noalign{\medskip}0&-2&0&0&1&1&-1&0&1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=1/2\,K_{{2}}-3/2\,K_{{1}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{1}}}^{2}+11\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=1
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=34
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&48
&-2
&0
&0
&0
\\ 
1 &2880
&45120
&135200
&45120
&2880
&0
\\ 
2 &208728
&31854624
&536009384
&2873844768
&5329853440
&2882233376
\\ 
3 &39767360
&21253338240
&1091380294400
&17654952181440
&123133222380800
&402894411622400
\\ 
4 &8984778408
&13561202128944
&1602986818855252
&56998790586691440
&893735526092204032
&7232461560670847280
\\ 
5 &2571518571840
&8412769444060608
&1940157518425228928
&129007680793510300992
&3749287422381954161792
&57492319634185857059520
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccccc} 1&0&1&1&0&0&0&0&0&0&1&1
\\\noalign{\medskip}0&1&0&0&1&1&1&1&1&1&1&1\\\noalign{\medskip}0&0&0&-
3&1&0&0&0&-1&-1&-2&-1\\\noalign{\medskip}0&0&0&-2&0&0&1&0&1&0&-1&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0&0&
-1&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&0&0
\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -1&0&0&0&0&0&1&0&0&-2&1&0
\\\noalign{\medskip}0&-1&-1&-1&0&0&0&-1&0&0&1&1\\\noalign{\medskip}0&0
&1&1&1&0&0&1&0&0&0&-2\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&1&-2&0
\\\noalign{\medskip}0&0&0&0&0&1&-1&0&1&-1&0&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{4}}
=D_{{8}}+2\,D_{{5}}+3\,D_{{4}}
$, $K_{{2}}
=D_{{8}}+2\,D_{{5}}+3\,D_{{4}}-D_{{1}}+D_{{3}}
$, $K_{{5}}
=D_{{4}}
$, $K_{{1}}
=D_{{5}}+D_{{4}}
$, $K_{{3}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{4}}D_{{7}},D_{{5}}D_{{9}},D_{{5}}D_
{{10}},D_{{8}}D_{{10}},D_{{9}}D_{{10}},D_{{1}}D_{{2}}D_{{3}}D_{{6}},D_
{{1}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{6}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{8}}
$, $D_{{9}}
$, $D_{{10}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccccc} -2&-6&1&1&3&0&2&1&0&0&0&0
\\\noalign{\medskip}0&-2&0&0&1&1&-1&0&1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=1/2\,K_{{2}}-3/2\,K_{{1}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{1}}}^{2}+11\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=1
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=34
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&48
&-2
&0
&0
&0
\\ 
1 &2880
&45120
&135200
&45120
&2880
&0
\\ 
2 &208728
&31854624
&536009384
&2873844768
&5329853440
&2882233376
\\ 
3 &39767360
&21253338240
&1091380294400
&17654952181440
&123133222380800
&402894411622400
\\ 
4 &8984778408
&13561202128944
&1602986818855252
&56998790586691440
&893735526092204032
&7232461560670847280
\\ 
5 &2571518571840
&8412769444060608
&1940157518425228928
&129007680793510300992
&3749287422381954161792
&57492319634185857059520
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 13 $}

\subsection{Partition $1$ of model $13$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} 1&0&0&1&1&0&0&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&1&1&0&0&1&1&1&1&1&1&2&2&2&3\\\noalign{\medskip}0
&0&0&-3&0&1&0&0&-1&-3&0&-1&-2&-1&-2\\\noalign{\medskip}0&0&0&-1&1&0&0&0
&0&-2&-1&-1&-1&-1&-2\\\noalign{\medskip}0&0&0&-1&0&0&0&1&0&-1&0&0&0&0&0
\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&-1&0&0&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&1&1&0&0&1\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} -1&0&0&0&1&0&0&0&-2&0&0&0&1&0&0
\\\noalign{\medskip}0&-1&0&0&0&0&-1&-1&0&-1&-1&0&0&1&1
\\\noalign{\medskip}0&0&1&0&0&0&0&0&0&0&0&-2&0&0&1\\\noalign{\medskip}0
&0&1&1&-1&0&0&0&0&-1&0&0&0&0&0\\\noalign{\medskip}0&0&-1&0&0&0&-1&-1&0
&-1&0&1&1&0&0\\\noalign{\medskip}0&0&0&0&0&1&1&1&0&1&0&0&0&-2&0
\\\noalign{\medskip}0&0&0&0&0&0&1&1&0&1&1&1&0&0&-2\\\noalign{\medskip}0
&0&0&0&0&0&1&1&1&1&0&0&-2&0&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{5}}
=-D_{{5}}+D_{{9}}+2\,D_{{2}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$, $K_{{2}}
=-D_{{5}}+D_{{7}}+D_{{10}}+2\,D_{{1}}+D_{{4}}+2\,D_{{3}}
$, $K_{{6}}
=D_{{4}}
$, $K_{{4}}
=D_{{2}}
$, $K_{{1}}
=D_{{3}}+D_{{2}}
$, $K_{{3}}
=D_{{1}}+D_{{2}}-D_{{5}}+D_{{9}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$, $K_{{7}}
=-D_{{5}}+D_{{9}}+D_{{10}}+2\,D_{{1}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$, $K_{{8}}
=D_{{7}}+2\,D_{{3}}+2\,D_{{2}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{1}}D_{{12}},D_{{1}}D_{{13}},D_{{2}}D_{{9}},D_{{2}}
D_{{10}},D_{{2}}D_{{13}},D_{{3}}D_{{9}},D_{{3}}D_{{10}},D_{{3}}D_{{11}
},D_{{3}}D_{{12}},D_{{3}}D_{{13}},D_{{4}}D_{{7}},D_{{4}}D_{{10}},D_{{4
}}D_{{11}},D_{{4}}D_{{13}},D_{{7}}D_{{9}},D_{{7}}D_{{10}},D_{{7}}D_{{
12}},D_{{7}}D_{{13}},D_{{9}}D_{{11}},D_{{10}}D_{{11}},D_{{10}}D_{{12}}
,D_{{11}}D_{{12}},D_{{11}}D_{{13}},D_{{12}}D_{{13}},D_{{3}}D_{{5}}D_{{
6}}D_{{8}},D_{{4}}D_{{5}}D_{{6}}D_{{8}},D_{{5}}D_{{6}}D_{{7}}D_{{8}},D
_{{1}}D_{{5}}D_{{6}}D_{{8}}D_{{9}},D_{{5}}D_{{6}}D_{{8}}D_{{9}}D_{{10}
}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{7}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$, $D_{{12}}
$, $D_{{13}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} -2&-6&0&0&2&3&1&1&0&1&0&0&0&0&0
\\\noalign{\medskip}0&0&1&1&-1&0&0&0&0&-1&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{4}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+3\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=3
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=42
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&4
&0
&0
&0
&0
\\ 
1 &2880
&4752
&252
&0
&0
&0
\\ 
2 &208728
&3603528
&2204928
&20520
&-9252
&0
\\ 
3 &39767360
&2445085664
&6905771456
&2493967312
&17671840
&-2695040
\\ 
4 &8984778408
&1580414726844
&12536489815488
&16303941022572
&4149078701904
&24150423420
\\ 
5 &2571518571840
&988726084819920
&17213021137437120
&54281784750359040
&43526767714418880
&8540389793334960
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} 1&0&0&1&1&0&0&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&1&1&0&0&1&1&1&1&1&1&2&2&2&3\\\noalign{\medskip}0
&0&0&-3&0&1&0&0&-1&-3&0&-1&-2&-1&-2\\\noalign{\medskip}0&0&0&-1&1&0&0&0
&0&-2&-1&-1&-1&-1&-2\\\noalign{\medskip}0&0&0&-1&0&0&0&1&0&-1&0&0&0&0&0
\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&-1&0&0&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0&1&1&0&0&1\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} -1&0&1&1&0&0&0&0&-2&-1&0&0&1&0
&0\\\noalign{\medskip}0&-1&0&0&0&0&-1&-1&0&-1&-1&0&0&1&1
\\\noalign{\medskip}0&0&-1&-1&1&0&0&0&0&1&0&0&0&0&0
\\\noalign{\medskip}0&0&1&0&0&0&0&0&0&0&0&-2&0&0&1\\\noalign{\medskip}0
&0&-1&0&0&0&-1&-1&0&-1&0&1&1&0&0\\\noalign{\medskip}0&0&0&0&0&1&1&1&0&
1&0&0&0&-2&0\\\noalign{\medskip}0&0&0&0&0&0&1&1&0&1&1&1&0&0&-2
\\\noalign{\medskip}0&0&0&0&0&0&1&1&1&1&0&0&-2&0&0\end {array}
 \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{6}}
=D_{{4}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{1}}
=D_{{2}}+D_{{3}}
$, $K_{{4}}
=-D_{{5}}+D_{{9}}+D_{{1}}+D_{{2}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$, $K_{{2}}
=-D_{{5}}+D_{{7}}+2\,D_{{1}}+D_{{10}}+D_{{4}}+2\,D_{{3}}
$, $K_{{8}}
=D_{{7}}+2\,D_{{2}}+2\,D_{{3}}
$, $K_{{5}}
=2\,D_{{2}}-D_{{5}}+D_{{9}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$, $K_{{7}}
=-D_{{5}}+D_{{9}}+2\,D_{{1}}+D_{{10}}+D_{{4}}+D_{{7}}+2\,D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{12}},D_{{1}}D_{{13}},D_{{2}}D_{{9}},D_{{2}}D_{{10}},D_{{2}
}D_{{13}},D_{{3}}D_{{8}},D_{{3}}D_{{9}},D_{{3}}D_{{10}},D_{{3}}D_{{11}
},D_{{3}}D_{{12}},D_{{3}}D_{{13}},D_{{4}}D_{{7}},D_{{4}}D_{{10}},D_{{4
}}D_{{11}},D_{{4}}D_{{13}},D_{{7}}D_{{9}},D_{{7}}D_{{10}},D_{{7}}D_{{
12}},D_{{7}}D_{{13}},D_{{9}}D_{{11}},D_{{10}}D_{{11}},D_{{10}}D_{{12}}
,D_{{11}}D_{{12}},D_{{11}}D_{{13}},D_{{12}}D_{{13}},D_{{1}}D_{{2}}D_{{
11}},D_{{1}}D_{{2}}D_{{5}}D_{{6}},D_{{4}}D_{{5}}D_{{6}}D_{{8}},D_{{5}}
D_{{6}}D_{{7}}D_{{8}},D_{{1}}D_{{5}}D_{{6}}D_{{8}}D_{{9}},D_{{5}}D_{{6
}}D_{{8}}D_{{9}}D_{{10}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{7}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$, $D_{{12}}
$, $D_{{13}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccccc} -2&-6&2&2&0&3&1&1&0&-1&0&0&0&0
&0\\\noalign{\medskip}0&0&-1&-1&1&0&0&0&0&1&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{3}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=10\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=3
&
K_{{1,1,2}}
&=4
&
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=42
&
\ch_2 \cdot J_2 &=52
&
\end{align}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&4
&0
&0
&0
&0
\\ 
1 &252
&4752
&2880
&0
&0
&0
\\ 
2 &-9252
&20520
&2204928
&3603528
&208728
&0
\\ 
3 &848628
&-2695040
&17671840
&2493967312
&6905771456
&2445085664
\\ 
4 &-114265008
&460972620
&-2654170560
&24150423420
&4149078701904
&16303941022572
\\ 
5 &18958064400
&-92258120448
&515459515968
&-3831306746112
&45162481474512
&8540389793334960
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $3$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} 1&0&0&1&1&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&1&1&0&0&1&1&1&1&1&1&2&2\\\noalign{\medskip}0&0&0
&-3&0&1&0&0&-1&-3&0&-2&-1\\\noalign{\medskip}0&0&0&-1&1&0&0&0&0&-2&-1&
-1&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&1&0&-1&0&0&0
\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0
&1&-1&0&0&0&0&0&0&1&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -1&0&-1&0&1&0&0&0&0&0&1&0&0
\\\noalign{\medskip}0&-1&0&0&0&0&-1&-1&0&-1&0&1&1\\\noalign{\medskip}0
&0&1&1&-1&0&0&0&0&-1&0&0&0\\\noalign{\medskip}0&0&1&0&0&0&0&0&-2&0&-1&
1&0\\\noalign{\medskip}0&0&0&0&0&1&1&1&0&1&0&0&-2\\\noalign{\medskip}0
&0&0&0&0&0&1&1&1&1&0&-2&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{3}}
=D_{{2}}
$, $K_{{5}}
=D_{{4}}
$, $K_{{1}}
=D_{{3}}+D_{{2}}
$, $K_{{4}}
=D_{{1}}+D_{{3}}
$, $K_{{2}}
=-D_{{5}}+2\,D_{{3}}+D_{{4}}+D_{{7}}+2\,D_{{1}}
$, $K_{{6}}
=D_{{7}}+2\,D_{{1}}+2\,D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{2}},D_{{1}}D_{{10}},D_{{1}}D_{{11}},D_{{2}}D_{{9}},D_{{3}}
D_{{9}},D_{{3}}D_{{10}},D_{{3}}D_{{11}},D_{{4}}D_{{7}},D_{{4}}D_{{10}}
,D_{{7}}D_{{11}},D_{{10}}D_{{11}},D_{{1}}D_{{5}}D_{{6}}D_{{8}},D_{{3}}
D_{{5}}D_{{6}}D_{{8}},D_{{4}}D_{{5}}D_{{6}}D_{{8}},D_{{5}}D_{{6}}D_{{7
}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{7}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -2&-6&0&0&2&3&1&1&0&1&0&0&0
\\\noalign{\medskip}0&0&1&1&-1&0&0&0&0&-1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{3}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+3\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=3
&
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=42
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&4
&0
&0
&0
&0
\\ 
1 &2880
&4752
&252
&0
&0
&0
\\ 
2 &208728
&3603528
&2204928
&20520
&-9252
&0
\\ 
3 &39767360
&2445085664
&6905771456
&2493967312
&17671840
&-2695040
\\ 
4 &8984778408
&1580414726844
&12536489815488
&16303941022572
&4149078701904
&24150423420
\\ 
5 &2571518571840
&988726084819920
&17213021137437120
&54281784750359040
&43526767714418880
&8540389793334960
\\ 
\hline
\end{array}
\end{equation}
This is triangulation $4$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} 1&0&0&1&1&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&1&1&0&0&1&1&1&1&1&1&2&2\\\noalign{\medskip}0&0&0
&-3&0&1&0&0&-1&-3&0&-2&-1\\\noalign{\medskip}0&0&0&-1&1&0&0&0&0&-2&-1&
-1&-1\\\noalign{\medskip}0&0&0&-1&0&0&0&1&0&-1&0&0&0
\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&-1&0&0&0\\\noalign{\medskip}0&0
&1&-1&0&0&0&0&0&0&1&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -1&0&0&1&0&0&0&0&0&-1&1&0&0
\\\noalign{\medskip}0&-1&0&0&0&0&-1&-1&0&-1&0&1&1\\\noalign{\medskip}0
&0&-1&-1&1&0&0&0&0&1&0&0&0\\\noalign{\medskip}0&0&1&0&0&0&0&0&-2&0&-1&
1&0\\\noalign{\medskip}0&0&0&0&0&1&1&1&0&1&0&0&-2\\\noalign{\medskip}0
&0&0&0&0&0&1&1&1&1&0&-2&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{2}}+D_{{3}}
$, $K_{{4}}
=D_{{1}}+D_{{3}}
$, $K_{{3}}
=D_{{3}}
$, $K_{{5}}
=D_{{4}}
$, $K_{{2}}
=-D_{{5}}+D_{{4}}+D_{{7}}+2\,D_{{1}}+2\,D_{{3}}
$, $K_{{6}}
=D_{{7}}+2\,D_{{1}}+2\,D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{1}}D_{{10}},D_{{1}}D_{{11}},D_{{2}}D_{{9}},D_{{3}}D_{{8}},D_{{3}}
D_{{9}},D_{{3}}D_{{10}},D_{{3}}D_{{11}},D_{{4}}D_{{7}},D_{{4}}D_{{10}}
,D_{{7}}D_{{11}},D_{{10}}D_{{11}},D_{{1}}D_{{2}}D_{{5}}D_{{6}},D_{{1}}
D_{{5}}D_{{6}}D_{{8}},D_{{4}}D_{{5}}D_{{6}}D_{{8}},D_{{5}}D_{{6}}D_{{7
}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 100 $.
The divisors $D_{{7}}
$, $D_{{9}}
$, $D_{{10}}
$, $D_{{11}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccccc} -2&-6&2&2&0&3&1&1&0&-1&0&0&0
\\\noalign{\medskip}0&0&-1&-1&1&0&0&0&0&1&0&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=1/2\,K_{{1}}
\qquad
J_{{2}}=K_{{3}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=10\,{J_{{1}}}^{2}-2\,{J_{{2}}}^{2}+5\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&=3
&
K_{{1,1,2}}
&=4
&
K_{{1,2,2}}
&=4
&
K_{{2,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=42
&
\ch_2 \cdot J_2 &=52
&
\end{align}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&4
&0
&0
&0
&0
\\ 
1 &252
&4752
&2880
&0
&0
&0
\\ 
2 &-9252
&20520
&2204928
&3603528
&208728
&0
\\ 
3 &848628
&-2695040
&17671840
&2493967312
&6905771456
&2445085664
\\ 
4 &-114265008
&460972620
&-2654170560
&24150423420
&4149078701904
&16303941022572
\\ 
5 &18958064400
&-92258120448
&515459515968
&-3831306746112
&45162481474512
&8540389793334960
\\ 
\hline
\end{array}
\end{equation}


\section{Polyhedron $ 1 $ in the list}

The model has $1$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&1&0&0&1&0&1
\\\noalign{\medskip}0&1&1&1&0&1&1&0&1&0\\\noalign{\medskip}0&0&0&3&0&-
1&0&0&1&0\\\noalign{\medskip}0&0&0&0&1&0&0&0&0&-1\\\noalign{\medskip}0
&0&0&-1&0&0&0&1&0&0\\\noalign{\medskip}0&0&0&-1&0&0&1&0&0&0
\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array} \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&1&0&0&0&0&1
\\\noalign{\medskip}-1&0&1&1&0&0&1&1&-3&0\\\noalign{\medskip}0&-2&0&0&0
&1&0&0&1&0\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{3}}
$, $K_{{2}}
=D_{{1}}
$, $K_{{3}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{4}}D_{{7}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,K_{{3}}K_{{2}}+4\,K_{{1}}K_{{3}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 128 $.
The divisors $D_{{7}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&0&0&0&1&0&0&0&0&1
\\\noalign{\medskip}-1&-6&1&1&0&3&1&1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{1}}
\qquad
J_{{2}}=K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)={J_{{1}}}^{2}+2\,J_{{2}}J_{{1}}+12\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=2
&
K_{{2,2,2}}
&=4
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=52
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&2496
&223752
&38637504
&9100224984
&2557481027520
\\ 
1 &2
&2496
&1941264
&1327392512
&861202986072
&540194037151104
\\ 
2 &0
&0
&223752
&1327392512
&2859010142112
&4247105405354496
\\ 
3 &0
&0
&0
&38637504
&861202986072
&4247105405354496
\\ 
4 &0
&0
&0
&0
&9100224984
&540194037151104
\\ 
5 &0
&0
&0
&0
&0
&2557481027520
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 3 $ in the list}

\subsection{Partition $1$ of model $3$}
The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccc} 1&0&0&1&0&0&0&1&0&0&1
\\\noalign{\medskip}0&1&1&0&1&1&1&0&1&1&1\\\noalign{\medskip}0&0&-1&0&
1&0&0&-3&0&-1&-2\\\noalign{\medskip}0&0&-1&0&0&0&0&-1&1&0&-1
\\\noalign{\medskip}0&0&1&0&0&1&0&-1&0&0&0\\\noalign{\medskip}0&0&0&0&0
&0&1&-1&0&0&0\\\noalign{\medskip}0&0&-1&1&0&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccc} -1&0&0&0&0&1&1&1&1&-3&0
\\\noalign{\medskip}0&-2&-1&-1&1&1&0&0&0&0&1\\\noalign{\medskip}0&0&1&
1&0&0&1&1&0&0&-2\\\noalign{\medskip}0&0&0&0&0&-1&-1&-1&0&1&1
\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{1}}
=D_{{7}}
$, $K_{{4}}
=-D_{{4}}+D_{{7}}+D_{{3}}
$, $K_{{3}}
=D_{{1}}+D_{{3}}
$, $K_{{2}}
=D_{{3}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{7}}D_{{9}},D_{{8}}D_{{9}},D_{{1}}D_{{2}}D_{{7}},D_
{{1}}D_{{2}}D_{{8}},D_{{3}}D_{{4}}D_{{9}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}
,D_{{3}}D_{{4}}D_{{5}}D_{{6}},D_{{4}}D_{{5}}D_{{6}}D_{{7}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 128 $.
The divisors $D_{{8}}
$, $D_{{9}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccc} -2&-12&0&0&6&2&2&2&2&0&0
\\\noalign{\medskip}-6&-24&3&3&12&0&3&3&6&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=3/2\,K_{{2}}-K_{{1}}
\qquad
J_{{2}}=-1/2\,K_{{2}}+1/2\,K_{{1}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=-58\,{J_{{2}}}^{2}-20\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,1}}
&={\frac {245}{2}}
&
K_{{1,1,2}}
&=-{\frac {63}{2}}
&
K_{{1,2,2}}
&=8
&
K_{{2,2,2}}
&=-2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=166
&
\ch_2 \cdot J_2 &=-44
&
\end{align}
This is triangulation $2$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{ccccccccccc} 1&0&0&1&0&0&0&1&0&0&1
\\\noalign{\medskip}0&1&1&0&1&1&1&0&1&1&1\\\noalign{\medskip}0&0&-1&0&
1&0&0&-3&0&-1&-2\\\noalign{\medskip}0&0&-1&0&0&0&0&-1&1&0&-1
\\\noalign{\medskip}0&0&1&0&0&1&0&-1&0&0&0\\\noalign{\medskip}0&0&0&0&0
&0&1&-1&0&0&0\\\noalign{\medskip}0&0&-1&1&0&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccc} -1&0&1&1&0&-1&0&0&1&-1&0
\\\noalign{\medskip}0&-2&0&0&1&0&0&0&0&1&0\\\noalign{\medskip}0&0&1&1&0
&0&1&1&0&0&-2\\\noalign{\medskip}0&0&-1&-1&0&1&0&0&0&-1&1\end {array}
 \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{3}}
=D_{{1}}+D_{{4}}
$, $K_{{1}}
=D_{{7}}
$, $K_{{2}}
=D_{{3}}
$, $K_{{4}}
=D_{{4}}+D_{{7}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{3}}D_{{8}},D_{{4}}D_{{9}},D_{{7}}D_{{9}},D_{{1}}D_{{2}}D_{{7}},D_
{{4}}D_{{5}}D_{{6}},D_{{1}}D_{{2}}D_{{5}}D_{{6}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,K_{{1}}K_{{2}}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 128 $.
The divisors $D_{{8}}
$, $D_{{9}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{ccccccccccc} -1&-2&1&1&1&-1&0&0&1&0&0
\\\noalign{\medskip}-1&-6&0&0&3&1&1&1&1&0&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=3/2\,K_{{1}}-1/2\,K_{{2}}
\qquad
J_{{2}}=-1/2\,K_{{1}}+1/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=12\,{J_{{2}}}^{2}+10\,J_{{2}}J_{{1}}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,2,2}}
&=2
&
K_{{2,2,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=24
&
\ch_2 \cdot J_2 &=44
&
\end{align}
$J_{{1}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&2496
&223752
&38637504
&9100224984
&2557481027520
\\ 
1 &24
&24000
&17244192
&11552340480
&7397501182992
&4599501889727232
\\ 
2 &-2
&4544
&92555600
&239152764672
&387426434941992
&494590812861015424
\\ 
3 &0
&-4096
&22945824
&911674812096
&4444162191527440
&12266354287567890432
\\ 
4 &0
&6144
&-12818168
&283693717248
&13387851546648736
&97181124180965503488
\\ 
5 &0
&-8192
&24969216
&-89080677888
&4975406530694672
&246059182807985648960
\\ 
\hline
\end{array}
\end{equation}

\section{Polyhedron $ 18 $ in the list}

The model has $2$ triangulations.
This is triangulation $1$.
The Gorenstein cone after desingularization is
\begin{equation}
 \left[ \begin {array}{cccccccccc} 1&0&0&0&1&0&1&0&0&0
\\\noalign{\medskip}0&1&1&1&0&1&0&1&1&1\\\noalign{\medskip}0&0&0&-3&0&0
&1&0&-2&-1\\\noalign{\medskip}0&0&0&-3&1&0&0&0&-2&-1
\\\noalign{\medskip}0&0&0&0&0&1&0&0&-1&0\\\noalign{\medskip}0&0&0&-1&0
&0&0&1&0&0\\\noalign{\medskip}0&0&1&-1&0&0&0&0&0&0\end {array}
 \right] 
\end{equation}
The Mori cone of the ambient space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -2&-1&0&0&1&0&1&0&0&1
\\\noalign{\medskip}0&0&1&1&0&0&0&1&0&-3\\\noalign{\medskip}0&0&0&0&0&
1&0&0&1&-2\end {array} \right] 
\end{equation}
A positive basis for the generators of the K\"ahler cone is $K_{{2}}
=D_{{1}}
$, $K_{{1}}
=D_{{3}}
$, $K_{{3}}
=D_{{4}}
$.
The Stanley Reisner ideal is
\begin{equation}
[D_{{4}}D_{{7}},D_{{1}}D_{{2}}D_{{6}},D_{{3}}D_{{5}}D_{{8}}]
\end{equation}
The Calabi-Yau space is defined by $
2\,{K_{{1}}}^{2}
$.
The Hodge numbers of this Calabi-Yau manifold are $h^{1,1}=2 $ and $h^{2,1}= 128 $.
The divisors $D_{{8}}
$ do not intersect the CY. 
The number of toric moduli is $\tilde{h}^{1,1} = 2$.
The Mori cone of the Calabi-Yau space is generated by
\begin{equation}
 \left[ \begin {array}{cccccccccc} -6&-3&1&1&3&0&3&1&0&0
\\\noalign{\medskip}-4&-2&0&0&2&1&2&0&1&0\end {array} \right] 
\end{equation}
The dual basis of divisors is
\begin{equation}
J_{{1}}=K_{{2}}
\qquad
J_{{2}}=1/2\,K_{{1}}-3/2\,K_{{2}}
\qquad
\end{equation}
Its second Chern class is
\begin{equation}
\ch_2(X)=18\,J_{{2}}J_{{1}}+12\,{J_{{1}}}^{2}+5\,{J_{{2}}}^{2}
\end{equation}
The non-zero intersection numbers are
\begin{align}
K_{{1,1,2}}
&=2
&
\end{align}
The linear forms are
\begin{align}
\ch_2 \cdot J_1 &=36
&
\ch_2 \cdot J_2 &=24
&
\end{align}
$J_{{1}}
$ is an elliptic surface.
$J_{{2}}
$ is a K3 surface with intersection form
\begin{equation}
 \left[ \begin {array}{c} 2\end {array} \right] 
\end{equation}
The genus zero instanton numbers up to degree 5 are
\begin{equation}
\begin{array}{|r|rrrrrr|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 \\ 
\hline
0 &0
&288
&252
&288
&252
&288
\\ 
1 &2496
&216576
&6391296
&104994816
&1209337344
&10917983232
\\ 
2 &223752
&152031744
&19638646848
&1180450842624
&43199009739072
&1107266933984256
\\ 
3 &38637504
&100021045248
&34832157566976
&4962537351009792
&401057938191181824
&21509207605173989376
\\ 
4 &9100224984
&63330228232704
&47042083144050624
&13025847457256417280
&1937385878589532624320
&184207098882508291501056
\\ 
5 &2557481027520
&39070309414708224
&53795625831623734272
&25710308960162910621696
&6317389886277581330494464
&962707308310765657585562112
\\ 
\hline
\end{array}
\end{equation}


\end{document}
