Coincidences between Calabi–Yau manifolds of Berglund–Hübsch type and Batyrev polytopes

We consider the phenomenon of the complete coincidence of key properties of Calabi–Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second is realized as a hypersurface in an orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and the same geometry on the complex structure moduli space and are also associated with the same $N=2$ gauged linear sigma model. We explain these coincidences using the correspondence between Calabi–Yau manifolds and the Batyrev reflexive polyhedra.