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 somewhat simpler.
We will do this on an example, and see how to obtain this information directly.
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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 $P^o$. 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).
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$P$ 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".
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$P_2$ 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".
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