Each 4D reflexive polytope in the KreuzerSkarke 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.



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 anticanonical divisor $X$ is a smooth CalabiYau 3fold 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).


$P$ 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".




$P_2$ 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".




