Families of Calabi-Yau hypersurfaces in $\mathbb Q$-Fano toric varieties

Abstract: We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-H\"ubsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes $\Delta_1\subseteq \Delta_2$, where $\Delta_1$ and $\Delta_2^*$ are canonical.