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-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 ${\mathrm{\Delta }}_1\subseteq {\mathrm{\Delta }}_2$, where ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2^{\ast }$ are canonical.