# Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

## Abstract

**:**

## 1. Introduction

## 2. Background

**Definition**

**1.**

**Theorem**

**1**

where ${S}_{\mathbf{w}}^{2n+1}$ denotes the unit sphere with a weighted Sasakian structure, $\mathbb{P}\left(\mathbf{w}\right)$ is weighted projective space coming from the quotient of ${S}_{\mathbf{w}}^{2n+1}$ by a weighted circle action generated from the weighted Sasakian structure. The top horizontal arrow is a Sasakian embedding and the bottom arrow is Kähler embedding. Moreover the vertical arrows are orbifold Riemannian submersions.

**Theorem**

**2**

## 3. Examples

(75,10,163,331,247) | z${}_{0}^{11}$+z${}_{0}{z}_{1}^{75}$+z${}_{1}{z}_{2}^{5}$+z${}_{2}{z}_{3}^{2}$+z${}_{3}{z}_{4}^{2}$ | 825 | $\mathbb{Z}$${}^{10}$⊕$\mathbb{Z}$${}_{55}$⊕ ($\mathbb{Z}$${}_{5}$)${}^{4}$ |

(62,124,155,9,85) | z${}_{0}^{7}$+z${}_{0}{z}_{1}^{3}$+z${}_{1}{z}_{2}^{2}$+z${}_{2}{z}_{3}^{31}$+z${}_{3}{z}_{4}^{5}$ | 434 | ${\mathbb{Z}}^{12}\oplus {\mathbb{Z}}_{14}\oplus $ (${\mathbb{Z}}_{2}$)${}^{2}$ |

(9,174,467,277,649) | z${}_{0}^{175}$+z${}_{0}{z}_{1}^{9}$+z${}_{1}{z}_{2}^{3}$+z${}_{2}{z}_{3}^{4}$+z${}_{3}{z}_{4}^{2}$ | 1575 | ${\mathbb{Z}}^{12}\oplus {\mathbb{Z}}_{525}\oplus $(${\mathbb{Z}}_{3}{)}^{2}$ |

(87,348,145,11,193) | z${}_{0}^{9}$+z${}_{0}{z}_{1}^{2}$+z${}_{1}{z}_{2}^{3}$+z${}_{2}{z}_{3}^{58}$+z${}_{3}{z}_{4}^{4}$ | 783 | ${\mathbb{Z}}^{12}\oplus {\mathbb{Z}}_{27}\oplus {\mathbb{Z}}_{3}$ |

(100,350,9,113,229) | z${}_{0}^{8}$+z${}_{0}{z}_{1}^{2}$+z${}_{1}{z}_{2}^{50}$+z${}_{2}{z}_{3}^{7}$+z${}_{3}{z}_{4}^{3}$ | 800 | ${\mathbb{Z}}^{14}\oplus {\mathbb{Z}}_{400}$ |

(9,291,488,181,787) | z${}_{0}^{195}$+z${}_{0}{z}_{1}^{6}$+z${}_{1}{z}_{2}^{3}$+z${}_{2}{z}_{3}^{7}$+z${}_{3}{z}_{4}^{2}$ | 1755 | ${\mathbb{Z}}^{14}\oplus {\mathbb{Z}}_{585}\oplus {\mathbb{Z}}_{3}$ |

(10,164,333,71,253) | z${}_{0}^{83}$+z${}_{0}{z}_{1}^{5}$+z${}_{1}{z}_{2}^{2}$+z${}_{2}{z}_{3}^{7}$+z${}_{3}{z}_{4}^{3}$ | 830 | ${\mathbb{Z}}^{14}\oplus {\mathbb{Z}}_{166}$ |

(10,540,275,163,103) | z${}_{0}^{109}$+z${}_{0}{z}_{1}^{2}$+z${}_{1}{z}_{2}^{2}$+z${}_{2}{z}_{3}^{5}$+z${}_{3}{z}_{4}^{9}$ | 1090 | ${\mathbb{Z}}^{16}\oplus {\mathbb{Z}}_{218}\oplus {\mathbb{Z}}_{2}$ |

(32,144,11,103,31) | z${}_{0}^{10}$+z${}_{0}{z}_{1}^{2}$+z${}_{1}{z}_{2}^{16}$+z${}_{2}{z}_{3}^{3}$+z${}_{3}{z}_{4}^{7}$ | 320 | ${\mathbb{Z}}^{18}\oplus {\mathbb{Z}}_{160}$ |

(45,36,27,11,107) | z${}_{0}^{5}$+z${}_{0}{z}_{1}^{5}$+z${}_{1}{z}_{2}^{7}$+z${}_{2}{z}_{3}^{18}$+z${}_{3}{z}_{4}^{2}$ | 225 | ${\mathbb{Z}}^{20}\oplus {\mathbb{Z}}_{5}$ |

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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Gomez, R.R. Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology. *Symmetry* **2019**, *11*, 947.
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Gomez RR. Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology. *Symmetry*. 2019; 11(7):947.
https://doi.org/10.3390/sym11070947

**Chicago/Turabian Style**

Gomez, Ralph R. 2019. "Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology" *Symmetry* 11, no. 7: 947.
https://doi.org/10.3390/sym11070947