Beglund-H\"ubsch transpose and Sasaki-Einstein rational homology 7-spheres
Abstract
We show that links of invertible polynomials coming from the Johnson and Koll\'ar list of K\"ahler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-H\"ubsch transpose rule coming from classical mirror symmetry constructions. Actually, this rule produces twins, that is, links with same degree, Milnor number and homology H3, with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number may vary. The Berglund-H\"ubsch transpose rule not only gives a framework to better understand the existence of SasakiEinstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7 -sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the BH-transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities.
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