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:
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 |