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ReflexivePolytopesDB :: Kreuzer-Skarke headers

Kreuzer-Skarke headers -- information contained in the header line

Each 4D reflexive polytope in the Kreuzer-Skarke database contains summary information about the polytope. Here, we explain this information.

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

str = getKreuzerSkarke(5,Limit=>1);
eg = first parseKS str;
A = matrixFromString eg_1
header = eg_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 Po of P
  • ’H: 5,20 [-30]’ are the Hodge numbers h1,1(X), h1,2(X), and the topological Euler characteristic of X, where X is the Calabi-Yau variety described next

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 Po. A fine regular star triangulation of Po defines a refined fan which corresponds 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 h1,1(X) = 5 and h1,2(X) = 20. The topological Euler characteristic of X is the number in square brackets: 2 h1,1(X) - 2 h1,2(X) = 10 - 40 = -30.

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

needsPackage "Polyhedra";
P = convexHull A

P is the convex hull of the columns in the M = ZZ4 lattice. P has 10 vertices and 25 lattice points, explaining the part of the line "M:25 10".

LP = latticePoints P
#LP
vertices P
numColumns vertices P
P2 = polar P
LP2 = latticePoints P2
#LP2
vertices P2
numColumns vertices P2

See also