Each 4D reflexive polytope in the Kreuzer-Skarke database contains summary information about the polytope. Here, we explain this information. A 3D polytope description line is similar, but somehwat simpler.

We will do this on an example, and see how to obtain this information directly.

i1 : topes = kreuzerSkarke(5,Limit=>1); using offline data file: ks5-n50.txt |

i2 : A = matrix topes_0 o2 = | 1 0 0 0 1 0 0 2 -2 -1 | | 0 1 0 0 0 1 -2 -1 2 0 | | 0 0 1 0 0 1 -1 -2 0 2 | | 0 0 0 1 1 0 2 0 -1 -2 | 4 10 o2 : Matrix ZZ <--- ZZ |

i3 : header = description topes_0 o3 = 4 10 M:25 10 N:10 9 H:5,20 [-30] id:0 |

This header line is what we wish to explain now.

The quick description:

- '4 10': the first 2 numbers are the number of rows and columns of the matrix $A$
- 'M:25 10': number of lattice points and the number of vertices of the 4-dimensional lattice polytope $P$ which is the convex hull of the columns of the matrix $A$
- 'N: 10 9' is the number of lattice points and the number of vertices of the polar dual polytope $P^o$ of $P$
- 'H: 5,20 [-30]' are the Hodge numbers $h^{1,1}(X)$, $h^{1,2}(X)$, and the topological Euler characteristic of $X$, where $X$ is the Calabi-Yau variety described next
- 'id: 0' is added by { t kreuzerSkarke}: each retrieved entry gets an id, starting with 0

Here, $X$ is defined as follows. Consider the Fano toric variety corresponding to the polytope $P$ (or, equivalently) to the fan determined by the polar dual polytope $P^o$. A fine regular star triangulation of $P^o$ defines a refined fan which corresponds to a simplicial toric variety $V$, such that a generic anti-canonical divisor $X$ is a smooth Calabi-Yau 3-fold hypersurface of $V$. The final numbers are about $X$: "H:5,20 [-30]" says that $h^{1,1}(X) = 5$ and $h^{1,2}(X) = 20$. The topological Euler characteristic of $X$ is the number in square brackets: $2 h^{1,1}(X) - 2 h^{1,2}(X) = 10 - 40 = -30$.

The first 2 integers are the dimensions of the matrix (4 by 10).

i4 : needsPackage "Polyhedra"; |

i5 : P = convexHull A o5 = P o5 : Polyhedron |

$P$ is the convex hull of the columns in the $M = \ZZ^4$ lattice. $P$ has 10 vertices and 25 lattice points, explaining the part of the line "M:25 10".

i6 : LP = latticePoints P o6 = {| -1 |, | 1 |, | 2 |, | 0 |, | 1 |, | -2 |, | 0 |, | 1 |, | 0 |, | | 0 | | 0 | | -1 | | 1 | | 0 | | 2 | | -2 | | 0 | | 1 | | | 2 | | 0 | | -2 | | 1 | | 0 | | 0 | | -1 | | -1 | | 0 | | | -2 | | 0 | | 0 | | 0 | | 1 | | -1 | | 2 | | 0 | | 0 | | ------------------------------------------------------------------------ 0 |, | -1 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | -1 |, 0, | -1 | | 1 | | -1 | | 0 | | 0 | | 0 | | -1 | | 0 | | 1 | | -1 | | 0 | | 0 | | 1 | | 0 | | 1 | | 0 | | -1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 1 | | -1 | | 0 | | 0 | | -1 | | ------------------------------------------------------------------------ 1 |, | 1 |, | -1 |, | -1 |, | -1 |, | 0 |} -1 | | -1 | | 0 | | 0 | | 1 | | 0 | -1 | | -1 | | 0 | | 1 | | 0 | | 0 | 1 | | 0 | | 0 | | -1 | | -1 | | -1 | o6 : List |

i7 : #LP o7 = 25 |

i8 : vertices P o8 = | 1 0 2 0 0 -1 -2 0 1 0 | | 0 1 -1 0 1 0 2 0 0 -2 | | 0 0 -2 1 1 2 0 0 0 -1 | | 0 0 0 0 0 -2 -1 1 1 2 | 4 10 o8 : Matrix QQ <--- QQ |

i9 : numColumns vertices P o9 = 10 |

$P_2$ is the polar dual of $P$ in the $N = \ZZ^4$ lattice. $P_2$ has 9 vertices and 10 lattice points, explaining the part of the line "N:10 9".

i10 : P2 = polar P o10 = P2 o10 : Polyhedron |

i11 : LP2 = latticePoints P2 o11 = {| 0 |, | -1 |, | 1 |, | -1 |, | 0 |, | -1 |, | 0 |, | 1 |, | -1 |, | 0 | | 0 | | 0 | | 1 | | -1 | | -1 | | -1 | | 1 | | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 1 | | 1 | | 0 | | -1 | | 0 | | -1 | | 0 | | -1 | | 0 | | -1 | | 1 | | 1 | ----------------------------------------------------------------------- 0} o11 : List |

i12 : #LP2 o12 = 10 |

i13 : vertices P2 o13 = | -1 -1 -1 -1 1 0 0 0 1 | | -1 0 1 -1 1 -1 -1 0 0 | | 0 -1 -1 0 1 0 1 -1 -1 | | 0 0 0 1 1 -1 -1 -1 -1 | 4 9 o13 : Matrix QQ <--- QQ |

i14 : numColumns vertices P2 o14 = 9 |

- kreuzerSkarke -- access Kreuzer-Skarke dimension 4 reflexive polytopes database
- KSEntry -- an entry from the Kreuzer-Skarke database of dimension 3 and 4 reflexive polytopes
- description(KSEntry) -- the description header