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

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-Hübsch-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-Hübsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes $Δ_1\subseteq Δ_2$, where $Δ_1$ and $Δ_2^*$ are canonical.