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Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f, mathbb{Z})$ is completely determined. Using the Boyer-Galicki construction of links $L_f$ over particular Kahler-Einstein orbifolds, we apply a valid case of Orliks conjecture to the links $L_f $ so that one is able to explicitly determine $H_{3}(L_f,mathbb{Z}).$ We give ten such new examples, all of which have the third Betti number satisfy $10leq b_{3}(L_{f})leq 20$.
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
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