I'm working with the Kreuzer-Skarke database of 4-dimensional reflexive polyhedra. It lists almost half a billion polytopes, each represented by its vertex list and a few properties (Hodge numbers and Euler's chi).
The database does not provide the structure of each polytope, i.e., the connectivity between vertices is not given. However, considering those polytopes are lattice-based and reflexive, I'd like to know whether it's possible to reconstruct the structure.
Does anyone have any ideas or strategies on how to accomplish this? Any help or pointers to relevant literature would be greatly appreciated.