Sasaki-Einstein rational homology spheres, rational varieties and the Berglund-Hübsch rule

We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hypersurfaces defined by cycle polynomials, on the estimate given by Johnson and Kollár to determine Kähler-Einstein orbifold metrics. We also construct weighted hypersurfaces that contain the rational varieties described above as codimension two subvarieties and, due to the refined estimate for cyclic polynomials, we find conditions on the weights and degrees of these hypersurfaces so their corresponding smooth links admit Sasaki-Einstein metrics. Finally we study the effect of the Berglund-Hübsch transpose rule on the topology and on the existence of Sasaki-Einstein metrics on the links studied and generalize all the results given in arXiv:2311.15998 for rational homology 7-spheres to rational homology $(4n-1)$-spheres, that is, we show invariance of these two features under the transpose rule.