Berglund–Hübsch transpose rule and Sasakian geometry

We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an \(n-1\)-dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension \(2n+1\) which are \(n-1\)-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four seven-dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein.