# Sasaki-Einstein 7-manifolds and Orlik’s conjecture

Jaime Cuadros Valle  and  Joe Lope Vicente  Departamento de Ciencias, Sección
Matemáticas, Pontificia Universidad Católica del Perú, Apartado 1761, Lima
100, Perú<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 2024)

###### Abstract.

We apply Orlik’s conjecture to study the homology groups of certain
2-connected 7-manifolds admitting quasi-regular Sasaki-Einstein metrics, among
them, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres,
extending the list given by Boyer, Galicki and Nakamaye in [7]. As a
consequence, we establish the existence of new families of homotopy 9-spheres
admitting positive Ricci curvature and determine the diffeomorphism type of
them. We also improve previous results given by Boyer [11] showing new
examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected
sums of $S^{3}\times S^{4}.$ Actually we show that manifolds of the form
$\\#k\left(S^{3}\times S^{4}\right)$ admit Sasaki-Einstein metrics for 22
different values of $k.$ All these links arise as Thom-Sebastiani sums of
chain type singularities and cycle type singularities where Orlik’s conjecture
holds due to a rather recent result by Hertling and Mase [17].

##  1\. Introduction

In the last 20 years, several techniques have been developed to determine the
class of manifolds that admit metrics of positive Ricci curvature. For odd
dimensions, Boyer, Galicki and collaborators established the abundance of
positive Einstein metrics, actually Sasaki-Einstein metrics. In dimension five
there are quite remarkable results (see [20], [9], [15]) that leads us to
think it is conceivable to give a complete classification there. For dimension
7, there are several important results, in particular 7-manifolds arising as
links of hypersurface singularities were studied intensively using as
framework the seminal work of Milnor in [21] where $S^{3}$-bundles over
$S^{4}$ with structure group $SO(4)$ were studied. Using the Gysin sequence
for these fiber bundles one obtains three classes of 7-manifolds determined by
the Euler class $e$:

  1. a)

If $e=\pm 1$ then the fiber bundle has the same homology of the 7-sphere.

  2. b)

If $|e|\geqslant 2$ then the fiber bundle is a rational homology 7-sphere.

  3. c)

If $e=0$ then the fiber bundle has the homology of $S^{3}\times S^{4}.$

Milnor showed that manifolds situated on the class described in item $a)$ were
homeomorphic to $S^{7}.$ Furthermore, he proved that some of these bundles are
not diffeomorphic to $S^{7}$ and as a result of that exhibited the first
examples of exotic spheres.

An interesting example of manifolds described in $b)$ is the Stiefel manifold
of 2-frames in $\mathbb{R}^{5}$, $V_{2}(\mathbb{R}^{5})$. It is known that
this manifold can be realized as a link of quadric in $\mathbb{C}^{5},$ and
that $V_{2}(\mathbb{R}^{5})$ is a rational homology 7-sphere and moreover
admits regular Sasaki-Einstein metric [3, 1]. This example was a source of new
techniques that led to establish the existence of many Sasaki-Einstein
structures. Actually in [3], a method for proving the existence of Sasaki-
Einstein metrics on links of hypersurface singularities is described. In a
sequel of papers [9], [6],[10], [11], Boyer, Galicki and their collaborators
showed the existence of Sasaki-Einstein structures on exotic spheres,
2-connected rational homology 7-spheres and connected sums of $S^{3}\times
S^{4}$ respectively.

In general, the homeomorphism type of links are not easy to be determined.
However, for certain cases, one can calculate the integral homology of the
links through formulas derived by Milnor and Orlik in [22] and Orlik in [23].
In fact, Boyer exhibited fourteen examples [11] of Sasaki-Einstein 7-manifolds
arising from links of isolated hypersurface singularities from elements of the
well-known list of 95 codimension one K3 surfaces ([18, 12]). For these, the
third integral homology group is completely determined. In [16] Gomez,
calculated the torsion for the third integral homology group explicitly for
ten examples of Sasaki-Einstein links of chain type singularities (also known
as Orlik polynomials).

In this paper we benefit from a recent result by Hertling and Mase in [17]
where they show that the Orlik conjecture is valid for chain type
singularities, cycle type singularities and Thom-Sebastiani sums of them, and
we update results given in [7], [11] and [16]. From the list of 1936 Sasaki-
Einstein 7-manifolds realized as links from the list given in [7, 19] we
detect 1673 that are links of hypersurface singularities of these types. Thus,
via Orlik’s algorithm we calculate the third homology group for this lot.
Among them, we found 52 new examples of 2-connected rational homology spheres
admitting Sasaki-Einstein metrics. We also found 124 new examples of
2-connected Sasaki-Einstein 7-manifolds of the form $\\#2k(S^{3}\times
S^{4})$, improving a result of Boyer’s in [11]. Six of these new examples are
links of quasi-smooth Fano 3-folds coming from Reids list of 95 weighted
codimension 1 K3 surfaces [18], [12], the rest of them, are links taken from
the list given in [19].

In recent years Sasaki-Einstein geometry has been intensely studied, part of
this interest comes from a string theory conjecture known as the AdS/CFT
correspondence which, under certain circumstances, relates superconformal
field theories and Sasaki-Einstein metrics in dimension five and seven [27].
Also, certain rational homology spheres can be used to construct positive
Sasakian structures in homotopy 9-spheres [8]. Thereby, it is important to
have as many examples as possible of Sasaki-Einstein manifolds especially in
dimensions five and seven.

This paper is organized as follows: in Section 2 we give some preliminaries on
the topology of links of hypersurface singularities. In Section 3, as a
consequence of the explicit calculations on the topology of new rational
Sasaki-Einstein homology 7-spheres given in this paper, we update some results
on homotopy 9-spheres admitting positive Ricci curvatures given in [8]. Then
we present Table 1 (listing new examples of rational homology 7-spheres
admitting Sasaki-Einstein metrics), Table 2 (listing new examples of 7-links
homeomorphic to connected sums of $S^{3}\times S^{4}$ admitting Sasaki-
Einstein metrics produced from the Johnson and Kollár list) and Table 3
(listing new examples of Sasaki-Einstein 7-links homeomorphic to connected
sums of $S^{3}\times S^{4}$ produced from Cheltsov’s list). In these three
tables we list the weights, one quasihomogenous polynomial generating the
link, the type of singularity, the degree, the Milnor number and finally the
third homology group. In Section 4, we give a link to the four codes
implemented in Matlab, these codes determine whether or not the links come
from the admissible type of singularities where Orlik’s conjecture is valid,
and compute the homology groups of the links under discussion. we also give
links to three additional tables (listing 7-links with non-zero third Betti
number and with torsion).

### Acknowledgments

The first author thanks Ralph Gomez and Charles Boyer for useful
conversations. Part of this article was prepared with the financial support
from Pontificia Universidad Católica del Perú through project DGI PI0655.

##  2\. Preliminaries: Sasaki-Einstein metrics on links and Orlik’s conjecture

In this section we briefly review the Sasakian geometry of links of isolated
hypersurface singularities defined by weighted homogeneous polynomials. We
describe the explicit constructions of Sasaki-Einstein manifolds given by
Boyer and Galicki [5]. Then we give some known facts on the topology of links
of hypersurface singularites [21], [22] and state Orlik’s conjecture. We also
set up a table with the necessary conditions to obtain links where this
conjecture is known to be valid.

###  2.1. Links and Sasaki-Einstein metrics

Consider the weighted $\mathbb{C}^{*}$ action on $\mathbb{C}^{n+1}$ given by

$\left(z_{0},\ldots,z_{n}\right)\longmapsto\left(\lambda^{w_{0}}z_{0},\ldots,\lambda^{w_{n}}z_{n}\right)$

where $w_{i}$ are the weights which are positive integers and
$\lambda\in\mathbb{C}^{*}$. Let us denote the weight vector by
$\mathbf{w}=\left(w_{0},\ldots,w_{n}\right).$ We assume
$\operatorname{gcd}\left(w_{0},\ldots,w_{n}\right)=1.$ Recall that a
polynomial $f\in\mathbb{C}\left[z_{0},\ldots,z_{n}\right]$ is said to be a
weighted homogeneous polynomial of degree $d$ and weight
$\mathbf{w}=\left(w_{0},\ldots,w_{n}\right)$ if for any
$\lambda\in\mathbb{C}^{*}=\mathbb{C}\backslash\\{0\\}$, we have

$f\left(\lambda^{w_{0}}z_{0},\ldots,\lambda^{w_{n}}z_{n}\right)=\lambda^{d}f\left(z_{0},\ldots,z_{n}\right).$

We are interested in those weighted homogeneous polynomials $f$ whose zero
locus in $\mathbb{C}^{n+1}$ has only an isolated singularity at the origin.
The link $L_{f}(\mathbf{w},d)$ (sometimes written $L_{f}$) is defined by

$L_{f}(\mathbf{w},d)=f^{-1}(0)\cap S^{2n+1},$

where $S^{2n+1}$ is the $(2n+1)$-sphere in $\mathbb{C}^{n+1}$. By the Milnor
fibration theorem [21], $L_{f}(\mathbf{w},d)$ is a closed $(n-2)$-connected
$(2n-1)$-manifold that bounds a parallelizable manifold with the homotopy type
of a bouquet of $n$-spheres. Furthermore, $L_{f}(\mathbf{w},d)$ admits a
quasi-regular Sasaki structure in a natural way, see for instance [26].
Moreover, if one considers the locally free $S^{1}$-action induced by the
weighted $\mathbb{C}^{*}$ action on $f^{-1}(0)$ the quotient space of the link
$L_{f}(\mathbf{w},d)$ by this action is the weighted hypersurface
$\mathcal{Z}_{f},$ a Kähler orbifold. Actually, we have the following
commutative diagram [5]

$\begin{CD}L_{f}(\mathbf{w},d)@>{{\qquad\qquad}}>{}>S^{2n+1}_{\bf w}\\\
@V{}V{\pi}V@V{}V{}V\\\
{\mathcal{Z}}_{f}@>{{\qquad\qquad}}>{}>{\mathbb{P}}({\bf w}),\end{CD}$

where $S_{\mathbf{w}}^{2n+1}$ denotes the unit sphere with a weighted Sasakian
structure, $\mathbb{P}(\mathbf{w})$ is a 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 Kählerian embedding and the vertical arrows
are orbifold Riemannian submersions.

It follows from the orbifold adjuntion formula that the link $L_{f}$ admits a
positive Ricci curvature if the quotient orbifold $Z_{f}$ by the natural
action $S^{1}$ is Fano, which is equivalent to

$|{\bf w}|-d_{f}>0,$ (1)

Here $|\mathbf{w}|=\sum_{i=0}^{m}w_{i}$ denotes the norm of the weight vector
$\mathbf{w}$ and $d_{f}$ is the degree of the polynomial $f$. Furthermore, in
[3], Boyer and Galicki found a method to obtain 2-connected Sasaki-Einstein
7-manifolds from the existence of orbifold Fano Kähler-Einstein hypersurfaces
$\mathcal{Z}_{f}$ in weighted projective 4-space $\mathbb{P}(\mathbf{w})$.
Actually, they showed a more general result:

######  Theorem 2.1.

The link $L_{f}(\mathbf{w},d)$ admits a Sasaki-Einstein structure if and only
if the Fano orbifold $\mathcal{Z}_{f}$ admits a Kähler-Einstein orbifold
metric of scalar curvature $4n(n+1).$

In [19], Johnson and Kollár give a list of 4442 quasi-smooth Fano 3 -folds
$\mathcal{Z}$ anticanonically embedded in weighted projective 4-spaces
$\mathbb{P}(\mathbf{w})$. Moreover, they show that 1936 of these 3 -folds
admit Kähler-Einstein metrics. Thus, such Fano 3-folds give rise to Sasaki-
Einstein metrics on smooth 7-manifolds realized as links of isolated
hypersurface singularities defined by weighted homogenous polynomials. In [7]
they extracted from this list 184 2-connected rational homology 7-spheres.
They also determined the order of
$H_{3}\left(L_{f}(\mathbf{w},d),\mathbb{Z}\right)$. In [16], Gomez used
Orlik’s conjecture to calculate the homology of 10 elements of the list given
in [7], all the 7-manifolds found there are links of chain type singularities.

In this paper we completely determine the third homology group for 1673 of
2-connected Sasaki-Einstein 7-manifolds from the list 1936 smooth 7-manifolds
realized as links of isolated hypersurface singularities from the list given
in [7]. Among them, we found 52 new examples of 2-connected rational homology
spheres admitting Sasaki-Einstein metrics. We also found 124 new examples of
2-connected Sasaki-Einstein 7-manifolds of the form $\\#2k(S^{3}\times
S^{4})$, improving a result of Boyer in [11], see Theorem 3.1. Of that lot,
118 come from the list given in [19], the other 6 examples are links of quasi-
smooth Fano 3-folds coming from Reids list of 95 weighted codimension 1 K3
surfaces, where all members of the list but 4 admit Kähler-Einstein metrics,
see [12].

###  2.2. The Topology of Links and Orlik’s Conjecture

In this section we review some classical results on the topology of links of
quasi-smooth hypersurface singularities. Recall that the Alexander polynomial
$\Delta_{f}(t)$ in [21] associated to a link $L_{f}$ of dimension $2n-1$ is
the characteristic polynomial of the monodromy map

$h_{*}:H_{n}(F,\mathbb{Z})\rightarrow H_{n}(F,\mathbb{Z})$

induced by the $S_{\mathrm{w}}^{1}$-action on the Milnor fibre $F$. Therefore,
$\Delta_{f}(t)=\operatorname{det}\left(t{\mathbb{I}}-h_{*}\right)$. Now both
$F$ and its closure $\bar{F}$ are homotopy equivalent to a bouquet of
$n$-spheres $S^{n}\vee\cdots\vee S^{n},$ and the boundary of $\bar{F}$ is the
link $L_{f}$, which is $(n-2)$-connected. The Betti numbers
$b_{n-1}\left(L_{f}\right)=b_{n}\left(L_{f}\right)$ equal the number of
factors of $(t-1)$ in $\Delta_{f}(t)$. The following are standard facts, see
[5].

  1. (1)

$L_{f}$ is a rational homology sphere if and only if $\Delta_{f}(1)\neq 0.$

  2. (2)

$L_{f}$ homotopy sphere if and only if $\Delta_{f}(1)=\pm 1.$

  3. (3)

If $L_{f}$ is a rational homology sphere, then the order of
$H_{n-1}\left(L_{f},\mathbb{Z}\right)$ equals $|\Delta(1)|.$

There is a remarkable theorem of Levine (see [21], page 69) that determines
the diffeomorphism type for rational homology spheres. More preceisely, we
have

######  Theorem 2.2.

Let $L_{f}$ be homeomorphic to the $(2n-1)$-sphere for $n$ odd. $L_{f}$ is
diffeomorphic to the standard sphere if $\Delta_{f}(-1)\equiv\pm 1(\bmod 8)$
and $L_{f}$ is diffeomorphic to the exotic Kervaire sphere if
$\Delta_{f}(-1)\equiv\pm 3(\bmod 8).$

In the case that $f$ is a weighted homogeneous polynomial there is an
algorithm due to Milnor and Orlik [22] to calculate the free part of
$H_{n-1}\left(L_{f},\mathbb{Z}\right).$ The authors associate to any monic
polynomial $f$ with roots $\alpha_{1},\ldots,\alpha_{k}\in\mathbb{C}^{*}$ its
divisor

$\operatorname{div}f=\left\langle\alpha_{1}\right\rangle+\cdots+\left\langle\alpha_{k}\right\rangle$

as an element of the integral ring $\mathbb{Z}\left[\mathbb{C}^{*}\right].$
Let $\Lambda_{n}=\operatorname{div}\left(t^{n}-1\right)$. Then the divisor of
$\Delta_{f}(t)$ is given by

$\operatorname{div}\Delta_{f}=\prod_{i=0}^{n}\left(\frac{\Lambda_{u_{i}}}{v_{i}}-1\right),$
(2)

where the $u_{i}^{\prime}s$ and $v_{i}^{\prime}s$ are given terms of the
degree $d$ of $f$ and the weight vector ${\bf w}=(w_{0},\ldots w_{n})$ by the
equations

$u_{i}=\frac{d}{\operatorname{gcd}\left(d,w_{i}\right)},\quad
v_{i}=\frac{w_{i}}{\operatorname{gcd}\left(d,w_{i}\right)}.$ (3)

Using the relations
$\Lambda_{a}\Lambda_{b}=\operatorname{gcd}(a,b)\Lambda_{\operatorname{lcm}(a,b)}$,
Equation (2) takes the form

$\operatorname{div}\Delta_{f}=(-1)^{n+1}+\sum a_{j}\Lambda_{j},$ (4)

where $a_{j}\in\mathbb{Z}$ and the sum is taken over the set of all least
common multiples of all combinations of the $u_{0},\ldots,u_{n}$. Then the
Alexander polynomial has an alternative expression given by

$\Delta_{f}(t)=(t-1)^{(-1)^{n}}\prod_{j}\left(t^{j}-1\right)^{a_{j}},$

and

$b_{n-1}\left(L_{f}\right)=(-1)^{n+1}+\sum_{j}a_{j}.$ (5)

Moreover, Milnor and Orlik gave and explicit formula to calculate the free
part of $H_{n-1}\left(L_{f},\mathbb{Z}\right):$

$b_{n-1}\left(L_{f}\right)=\sum(-1)^{n+1-s}\frac{u_{i_{1}}\cdots
u_{i_{s}}}{v_{i_{1}}\cdots
v_{i_{s}}\operatorname{lcm}\left(u_{i_{1}},\ldots,u_{i_{s}}\right)},$ (6)

where the sum is taken over all the $2^{n+1}$ subsets
$\left\\{i_{1},\ldots,i_{s}\right\\}$ of $\\{0,\ldots,n\\}$. In [23], Orlik
gave a conjecture which allows to determine the torsion of homology of the
link solely in terms of the weight of $f.$

######  Conjecture 2.3 (Orlik).

Consider $\left\\{i_{1},\ldots,i_{s}\right\\}\subset\\{0,1,\ldots,n\\}$ the
set of ordered set of $s$ indices, that is, $i_{1}<i_{2}<\cdots<i_{s}$. Let us
denote by I its power set (consisting of all of the $2^{s}$ subsets of the
set), and by $J$ the set of all proper subsets. Given a $(2n+2)$-tuple
$(\mathbf{u},\mathbf{v})=\left(u_{0},\ldots,u_{n},v_{0},\ldots,v_{n}\right)$
of integers, let us define inductively a set of $2^{s}$ positive integers, one
for each ordered element of $I$, as follows:

$c_{\emptyset}=\operatorname{gcd}\left(u_{0},\ldots,u_{n}\right),$

and if $\left\\{i_{1},\ldots,i_{s}\right\\}\in I$ is ordered, then

$c_{i_{1},\ldots,i_{s}}=\frac{\operatorname{gcd}\left(u_{0},\ldots,\hat{u}_{i_{1}},\ldots,\hat{u}_{i_{s}},\ldots,u_{n}\right)}{\prod_{J}c_{j_{1},\ldots
j_{t}}}.$

Similarly, we also define a set of $2^{s}$ real numbers by

$k_{\emptyset}=\epsilon_{n+1},$

and

$k_{i_{1},\ldots,i_{s}}=\epsilon_{n-s+1}\sum_{I}(-1)^{s-t}\frac{u_{j_{1}}\cdots
u_{j_{t}}}{v_{j_{1}}\cdots
v_{j_{t}}\operatorname{lcm}\left(u_{j_{1}},\ldots,u_{j_{t}}\right)}$

where

$\epsilon_{n-s+1}=\begin{cases}0&\text{ if }n-s+1\text{ is even }\\\ 1&\text{
if }n-s+1\text{ is odd }\end{cases}$

respectively. Finally, for any $j$ such that $1\leqslant j\leqslant
r=\left\lfloor\max\left\\{k_{i_{1},\ldots,i_{s}}\right\\}\right\rfloor$, where
$\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, we set
$d_{j}=\prod_{k_{i_{1},\ldots,i_{s}}\geqslant j}c_{i_{1},\ldots,i_{s}}.$ Then

$H_{n-1}\left(L_{f},\mathbb{Z}\right)_{\text{tor
}}=\mathbb{Z}/d_{1}\oplus\cdots\oplus\mathbb{Z}/d_{r}.$ (7)

This conjecture was known to hold in certain special cases [24]. In a rather
recent paper, Hertling and Mase [17], extend the list of possible cases and
showed that this conjecture is true for the following cases:

  1. (1)

Chain type singularity: that is, a quasihomogeneous singularity of the form

$f=f\left(x_{1},\ldots,x_{n}\right)=x_{1}^{a_{1}+1}+\sum_{i=2}^{n}x_{i-1}x_{i}^{a_{i}}$

for some $n\in\mathbb{N}$ and some $a_{1},\ldots,a_{n}\in\mathbb{N}$.

  2. (2)

Cycle type singularity: a quasihomogeneous singularity of the form

$f=f\left(x_{1},\ldots,x_{n}\right)=\sum_{i=1}^{n-1}x_{i}^{a_{i}}x_{i+1}+x_{n}^{a_{n}}x_{1}$

for some $n\in\mathbb{Z}_{\geqslant 2}$ and some
$a_{1},\ldots,a_{n}\in\mathbb{N}$ which satisfy for even $n$ neither $a_{j}=1$
for all even $j$ nor $a_{j}=1$ for all odd $j.$

  3. (3)

Thom-Sebastiani iterated sums of singularties of chain type or cyclic type.
Recall, for $f$ and $g$ be singularities, the Thom-Sebastiani sum is given by

$f+g=f\left(x_{1},\ldots,x_{n_{f}}\right)+g\left(x_{n_{f}+1},\ldots,x_{n_{f}+n_{g}}\right).$

Any iterated Thom-Sebastiani sum of chain type singularities and cycle type
singularities are also called invertible polynomials.

  4. (4)

Although Brieskorn-Pham singularities, or BP singularities

$f=f\left(x_{1},\ldots,x_{n}\right)=\sum_{i=1}^{n}x_{i}^{a_{i}}$

for some $n\in\mathbb{N}$ and some $a_{1},\ldots,a_{n}\in\mathbb{Z}_{\geqslant
2}$ are special cases of chain type singularities we prefer to label them as
an independent type of singularity.

In order to use Orlik’s conjecture, we need to find from the two lists of
Kähler-Einstein oribifolds aforementioned, elements with weights
$\mathbf{w}=\left(w_{0},w_{1},w_{2},w_{3},w_{4}\right)$ and degree
$d=\left(w_{0}+\cdots+w_{4}\right)-1$ that can be represented by BP
singularities, chain type singularities, cycle type singularities or Thom
Sebastiani sum of these types of singularities. Thus, given a weight vector
$\mathbf{w}=\left(w_{0},w_{1},w_{2},w_{3},w_{4}\right)$ one needs to determine
if there exist exponents $a_{i}$’s that verify certain arithmetic condition.
We include these conditions in the table below.

Most of the examples that we describe in this article have large weights and
degrees, so in order to avoid monotonous calculations, codes were implemented
in Matlab (see $(d)$ in Appendix) to determine for a given weight vector
$\mathbf{w}=\left(w_{0},w_{1},w_{2},w_{3},w_{4}\right)$ the exponents
$a_{i}$’s, such that the singularity can be written as a chain type, cycle
type or Thom-Sebastiani sum of them. We also wrote a code to computes the
Betti numbers and the numbers $d_{i}$ which generate the torsion in
$H_{3}\left(L_{f},\mathbb{Z}\right)$.

Type |  Polynomial |  Condition
---|---|---
BP |  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{2}^{a_{2}}+z_{3}^{a_{3}}+z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=a_{2}w_{2}=a_{3}w_{3}=a_{4}w_{4}$
Chain |  $z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{0}+a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
Cycle |  $z_{4}z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=w_{4}+a_{0}w_{0}=w_{0}+a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
|  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
BP + Chain |  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
|  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{2}^{a_{2}}+z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=a_{2}w_{2}=a_{3}w_{3}=w_{3}+a_{4}w_{4}$
|  $z_{0}^{a_{0}}+z_{4}z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{4}+a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
BP + Cycle |  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{4}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=w_{4}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
|  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{2}^{a_{2}}+z_{4}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=a_{2}w_{2}=w_{4}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
Chain + Chain |  $z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{0}+a_{1}w_{1}=a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
Chain + Cycle |  $z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{4}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{0}+a_{1}w_{1}=w_{4}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
|  $z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{4}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{0}+a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{4}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
Cycle + Cycle |  $z_{1}z_{0}^{a_{0}}+z_{0}z_{1}^{a_{1}}+z_{4}z_{2}^{a_{2}}+z_{2}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=w_{1}+a_{0}w_{0}=w_{0}+a_{1}w_{1}=w_{4}+a_{2}w_{2}=w_{2}+a_{3}w_{3}=w_{3}+a_{4}w_{4}$
BP + Chain + Chain |  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=w_{1}+a_{2}w_{2}=a_{3}w_{3}=w_{4}+w_{3}+a_{4}w_{4}$
BP + Chain + Cycle |  $z_{0}^{a_{0}}+z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{4}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{4}+a_{3}w_{3}=w_{4}+w_{3}+a_{4}w_{4}$
BP + Cycle + Cycle |  $z_{0}^{a_{0}}+z_{2}z_{1}^{a_{1}}+z_{1}z_{2}^{a_{2}}+z_{4}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ |  $d=a_{0}w_{0}=w_{2}+a_{1}w_{1}=w_{1}+a_{2}w_{2}=w_{4}+a_{3}w_{3}=w_{4}+w_{3}+a_{4}w_{4}$

Next, we discuss an interesting result on the topology of the link if the
degree $d$ and the weight vector ${\bf w}$ are such that
$\operatorname{gcd}(d,w_{i})=1$ for all $i.$ Several elements on the tables we
present in Section 3, satisfy this condition. Notice that, if the type of
singularities is restricted to the ten cases described on the table given
above, then $\operatorname{gcd}(d,w_{i})=1$ for all $i$ forces the singularity
to be of cycle type. We restrict to dimension 7, but this remark can be easily
generalized for any dimension.

######  Lemma 2.4.

Consider a 7-manifold $M$ arising as a link of a hypersurface singularity with
degree $d$ and such that $gcd(d,w_{i})=1$ for all $i=0,\dots 4$, then
$\mu+1=d(b_{3}+1)$ and $H_{3}(M,\mathbb{Z})_{tor}=\mathbb{Z}_{d}.$ In
particular, for hypersurface singularities that determine a rational homology
sphere with $\operatorname{gcd}\left(d,w_{i}\right)=1$ for all $i$ we obtain
exactly $d-1$ vanishing cycles.

###### Proof.

Indeed, from Equation $(6)$ we have

$b_{3}=\sum(-1)^{5-s}\left[\dfrac{u_{i_{1}}\dots u_{i_{s}}}{v_{i_{1}}\dots
v_{i_{s}}lcm(u_{i_{1}}\dots u_{i_{s}})}\right],$

where the sum is over all subsets $\\{i_{1}\dots
i_{s}\\}\subset\\{0,1,2,3,4\\}$ with $i_{1}<\dots<i_{s}$ and

$u_{i}=\dfrac{d}{gcd(d,w_{i})},v_{i}=\dfrac{w_{i}}{gcd(d,w_{i})},\forall
i=0,\dots,4.$

Since $gcd(d,w_{i})=1$ we have $u_{i}=d$ and $v_{i}=w_{i}$. Then the above
formula can be simplified as follows

$b_{3}=-1+\sum\dfrac{1}{w_{i_{1}}}-\sum\dfrac{d}{w_{i_{1}}w_{i_{2}}}+\sum\dfrac{d^{2}}{w_{i_{1}}w_{i_{2}}w_{i_{3}}}-\sum\dfrac{d^{3}}{w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}}}+\dfrac{d^{4}}{w_{0}w_{1}w_{2}w_{3}w_{4}}.$

Writing $S_{1}=\sum w_{i_{1}},\ \ S_{2}=\sum w_{i_{1}}w_{i_{2}},\ \ S_{3}=\sum
w_{i_{1}}w_{i_{2}}w_{i_{3}},\ \ S_{4}=\sum
w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}},$ we obtain

$b_{3}=-1+\dfrac{S_{4}}{w_{0}\dots w_{4}}-\dfrac{S_{3}d}{w_{0}\dots
w_{4}}+\dfrac{S_{2}d^{2}}{w_{0}\dots w_{4}}-\dfrac{S_{1}d^{3}}{w_{0}\dots
w_{4}}+\dfrac{d^{4}}{w_{0}\dots w_{4}}$ (8)

On the other hand, the Milnor number is given by
$\mu=\left(\dfrac{d}{w_{0}}-1\right)\dots\left(\dfrac{d}{w_{4}}-1\right).$
Then

$\mu=\dfrac{d^{5}-S_{1}d^{4}+S_{2}d^{3}-S_{3}d^{2}+S_{4}d-w_{0}\dots
w_{4}}{w_{0}\dots w_{4}}.$

From (8), we obtain $\mu+1=d(b_{3}+1).$ For the second claim, one notices that
$gcd(d,w_{i})=1$ implies $u_{i}=d$ and $v_{i}=w_{i}$, for all $i$. Thus
$c_{\emptyset}=gcd(d,\dots,d)=d\ \ \mbox{and}\ \ \ c_{i_{1}\dots i_{s}}=1.$
Clearly $k_{\emptyset}=1$. Then, for $j=1$, we get $d_{1}=d$, while in other
cases, $d_{j}=1$. Therefore, from Equation $(7)$ one concludes that if
$\operatorname{gcd}(d,w_{i})=1$ for all $i=0,\dots n$ then
$H_{3}(M,\mathbb{Z})_{tor}=\mathbb{Z}_{d}.$ ∎

##  3\. Results

In this section we present three tables of new examples of two classes of
7-manifolds: rational homology 7-spheres, and 7-manifolds homeomorphic to
connected sums of $S^{3}\times S^{4}$, all admitting Sasaki-Einstein metrics.
These new examples are extracted from two lists of Kähler-Einstein orbifolds,
the first one given by Johnson and Kollár in [19] (see $(a)$ in Appendix for
link to the list) and the other one given by Cheltsov in [12]. In these three
tables we list the weights, the quasihomogenous polynomial generating the
link, the type of singularity, the degree, the Milnor number and finally the
third homology group. But, first, we discuss the diffeomorphism type of
certain homotopy spheres admitting positive Ricci curvature.

###  3.1. Positive Ricci curvature on homotopy spheres

The existence of positive Sasakian metrics on homotopy spheres was study in
detail by Boyer, Galicki and collaborators, where the authors exhibited
inequivalent families of homotopy spheres admitting this type of metrics based
on their list of 184 rational homology 7-spheres admitting Sasakian-Einstein
metrics. We apply the methods developed in [7] and [8] to the list of the new
52 rational homology 7-spheres presented in this paper. Let us briefly review
some basic facts on links viewed as branched covers. The main reference here
is [8].

Let $f=f\left(z_{1},\ldots,z_{m}\right)$ be a quasi-smooth weighted
homogeneous polynomial of degree $d_{f}$ in $m$ complex variables, and let
$L_{f}$ denote its link. Let $\mathbf{w}_{f}=\left(w_{1},\ldots,w_{m}\right)$
be the corresponding weight vector. We consider branched covers constructed as
the link $L_{g}$ of the polynomial

$g=z_{0}^{p}+f\left(z_{1},\ldots,z_{m}\right),$

for $p>1.$ Then $L_{g}$ is a $p$-fold branched cover of $S^{2m+1}$ branched
over the link $L_{f}$. The degree of $L_{g}$ is
$d_{g}=\operatorname{lcm}\left(p,d_{f}\right)$, and the weight vector is
$\mathbf{w}_{g}=\left(\frac{d_{f}}{\operatorname{gcd}\left(p,d_{f}\right)},\frac{p}{\operatorname{gcd}\left(p,d_{f}\right)}\mathbf{w}_{f}\right)$.

We have

$|{\bf w}_{g}|-d_{g}=\frac{d_{f}+p(|{\bf
w}_{f}|-d_{f})}{\operatorname{gcd}\left(p,d_{f}\right)}.$ (9)

Thus, from Equation (1) and Theorem 2.1 it follows that $L_{g}$ admits
positive Ricci curvature for all $p>0$ if $L_{f}$ admits a Sasaki-Einstein
metric.

Furthermore, in [8] the authors showed that the link $L_{g}$ is a homotopy
sphere if and only if $L_{f}$ is a rational homology sphere. Let us recall the
argument, since it will be useful for our purpose. Here we assume that
$\operatorname{gcd}\left(p,d_{f}\right)=1.$

For the divisor $\operatorname{div}\Delta_{g}$ we have the equalities

$\displaystyle\operatorname{div}\Delta_{g}$
$\displaystyle=\left(\Lambda_{p}-1\right)\operatorname{div}\Delta_{f}=\left(\Lambda_{p}-1\right)\left((-1)^{n}+\sum
a_{j}\Lambda_{j}\right)$
$\displaystyle=\sum_{j}\operatorname{gcd}(p,j)a_{j}\Lambda_{\operatorname{lcm}(p,j)}-\sum_{j}a_{j}\Lambda_{j}+(-1)^{n}\Lambda_{p}+(-1)^{n+1}.$

Since the $j$ ’s run through all the least common multiples of the set
$\left\\{u_{1},\ldots,u_{n}\right\\}$ and
$\operatorname{gcd}\left(p,u_{i}\right)=1$ for all $i$, we note that for all
$j,\operatorname{gcd}(p,j)=1$. Thus

$b_{n-1}\left(L_{g}\right)=\sum_{j}a_{j}-\sum_{j}a_{j}+(-1)^{n}+(-1)^{n+1}=0.$

So $L_{g}$ is a rational homology sphere. Now, the computation of the
Alexander polynomial for $L_{g}$ leads to:

$\displaystyle\Delta_{g}(t)$
$\displaystyle=(t-1)^{(-1)^{n+1}}\left(t^{p}-1\right)^{(-1)^{n}}\prod_{j}\left(t^{pj}-1\right)^{a_{j}}\left(t^{j}-1\right)^{-a_{j}}$
$\displaystyle=\left(t^{p-1}+\cdots+t+1\right)^{(-1)^{n}}\prod_{j}\left(\frac{t^{pj-1}+\cdots+t+1}{t^{j-1}+\cdots+t+1}\right)^{a_{j}}$
$\displaystyle=\left(t^{p-1}+\cdots+t+1\right)^{(-1)^{n}}\prod_{j}\left(t^{(p-1)j}+\cdots+t^{j}+1\right)^{a_{j}}.$
(10)

This gives

$\Delta_{g}(1)=p^{\Sigma_{j}a_{j}+(-1)^{n}}.$

Thus from Equation (5) it follows that $L_{g}$ is a homotopy sphere if and
only if $L_{f}$ is a rational homology sphere.

In [8], the authors made use of Equation (10) and apply Theorem 2.2 to
determine the diffeomorphism type for $L_{g}$ for 7-dimensional links $L_{f}$
coming from their list of rational homology 7-spheres [7]. If $L_{f}$ has odd
degree they proved.

######  Theorem 3.1 (Boyer et al. [7]).

Let $L_{g}$ be the link of a $p$-branched cover of $S^{2n+1}$ branched over a
link $L_{f}$ in their list of 184 rational homology spheres given in [7].
Suppose that degree $d_{f}$ of $L_{f}$ is odd and that
$\operatorname{gcd}(p,d_{f})=1.$ Then, for $p$ odd, $L_{g}$ is diffeomorphic
to the standard 9-sphere $S^{9}.$ For $p$ even, $L_{g}$ is diffeomorphic to
$S^{9}$ if $|H_{3}(L_{f},\mathbb{Z})|\equiv\pm 1(\bmod 8)$ and $L_{g}$ is
diffeomorphic to the Kervaire exotic sphere $\Sigma_{9}$ if
$|H_{3}(L_{f},\mathbb{Z})|\equiv\pm 3(\bmod 8).$

The next theorem is an update of Theorems 7.6 and Theorems 7.7 in [8] and its
proof is similar to the one given by Boyer et al.

######  Theorem 3.2.

Let us assume for $\operatorname{gcd}(p,d_{f})=1$, then for each member of the
52 rational homology spheres in our list given in Table 1, $L_{g}$ is a
homotopy sphere admitting positive Ricci curvature. If $L_{f}$ is taken to be
one of the 49 rational homology spheres with odd degree listed in Table 1,
then $L_{g}$ admits Sasakian metrics with positive Ricci curvature and these
are diffeomorphic to the standard sphere $S^{9}.$ For the remaining three
members with even degree in Table 1, we have

  1. (1)

For $L_{f}$ with ${\bf w}=(118,118,185,135,35)$ with degree $d_{f}=590,$
$L_{g}$ diffeomorphic to the standard $S^{9}$ if $p^{4}\equiv\pm 1(\bmod 8)$
and diffeomorphic to the exotic Kervaire 9-sphere $\Sigma_{9}$ if
$p^{4}\equiv\pm 3(\bmod 8).$

  2. (2)

For $L_{f}$ with ${\bf w}=(64,512,475,375,175)$ with degree $d_{f}=1600,$
$L_{g}$ diffeomorphic to the standard $S^{9}$ if $p^{3}\equiv\pm 1(\bmod 8)$
and diffeomorphic to the exotic Kervaire 9-sphere $\Sigma_{9}$ if
$p^{3}\equiv\pm 3(\bmod 8).$

  3. (3)

For $L_{f}$ with ${\bf w}=(3532,7064,5355,115,1595)$ with degree
$d_{f}=17660,$ $L_{g}$ diffeomorphic to the standard $S^{9}$ if
$p^{2}\equiv\pm 1(\bmod 8)$ and diffeomorphic to the exotic Kervaire 9-sphere
$\Sigma_{9}$ if $p^{2}\equiv\pm 3(\bmod 8).$

###### Proof.

Our list of rational homology 7-spheres admitting Sasaki-Einstein metrics
contains 49 elements with odd degree and all of these have order
$\left|H_{3}\right|\equiv 1(\bmod 8),$ thus, the first part of the theorem
follows from Theorem 3.1. For the three members with even degree in Table 1,
the weight vectors can be written as
$\textbf{w}=(w_{0},w_{1},w_{2},w_{3},w_{4})=(m_{2}v_{0},m_{2}v_{1},m_{2}v_{2},m_{3}v_{3},m_{3}v_{4})$
where $\gcd(m_{2},m_{3})=1$ and $m_{2}m_{3}=d$ where $m_{2}$ is odd and
$m_{3}$ is even. For these one obtains

$\operatorname{div}\Delta_{f}=n({\bf w})\Lambda_{d}+\Lambda_{m_{3}}-n({\bf
w})\Lambda_{m_{2}}-1,$ (11)

where $n({\bf w})$ is a positive integer. It follows from Equation (4) that
$\sum_{j\,even}a_{j}=n({\bf w})+1$.

On the other hand, bearing in mind Theorem 2.2, we pay attention to
$\Delta_{g}(-1):$ since $p$ is odd and $d$ even, from Equation (10) we have

$\Delta_{g}(-1)=\prod_{j\text{ even }}p^{a_{j}}=p^{\Sigma_{j\text{ even
}}^{a_{j}}}$

Since $L_{f}$ is a rational homology sphere
$H_{3}(L_{f},\mathbb{Z})_{tor}=\Delta(1),$ so in order to compute $\Delta(1)$
we rewrite the equation (11) in terms of the Alexander polynomial:

$\displaystyle\Delta(t)$ $\displaystyle=$ $\displaystyle(t^{d}-1)^{n({\bf
w})}(t^{m_{3}}-1)(t^{m_{2}}-1)^{-n({\bf w})}(t-1)^{-1}$ $\displaystyle=$
$\displaystyle(t^{d-1}+\ldots 1)^{n({\bf w})}(t^{m_{3}-1}+\ldots
1)(t^{m_{2}-1}+\ldots 1)^{-n({\bf w})},$

Thus $\Delta(1)=dm_{3}m_{2}^{-n({\bf w})}=m_{3}^{n({\bf w})+1}.$ The result
follows by checking the torsion of $H_{3}(L_{f},\mathbb{Z})$ at Table 1 below.

∎

###  3.2. Rational homology 7-spheres admitting Sasaki-Einstein structures
from Johnson-Kollár list

In [19] Johnson and Kollár gave a list of 4442 of well-formed quasismooth
$\mathbb{Q}$-Fano 3-folds of index one anti-canonically embedded in
$\mathbb{CP}^{4}(\mathbf{w}).$ This list contains 1936 3-folds that admit a
Kähler-Einstein metric. Theorem 2.1 implies that the corresponding link admits
Sasaki-Einstein metrics. These links were studied in [7] and they found 184
rational homology 7-spheres. In the following table, we present 52 new
examples of 2-connected rational homology spheres admitting Sasaki-Einstein
metrics.

Table 1: Rational homology 7-spheres admitting S-E structures

${\bf{w}}=(w_{0},w_{1},w_{2},w_{3},w_{4})$ | Polynomial | Type | $d$ | $\mu$ | $H_{3}(M,\mathbb{Z})$
---|---|---|---|---|---
(13,143,775,620,465) | $z_{0}^{155}+z_{0}z_{1}^{14}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{3}$ | Chain + Cycle | 2015 | 24192 | $(\mathbb{Z}_{13})^{14}$
(77,77,333,180,27) | $z_{0}^{9}+z_{1}^{9}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{19}$ | BP + Cycle | 693 | 4864 | $(\mathbb{Z}_{77})^{8}$
(67,67,161,28,147) | $z_{0}^{7}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{11}+z_{3}z_{4}^{3}$ | BP + Cycle | 469 | 2376 | $(\mathbb{Z}_{67})^{6}$
(29,667,1807,1112,417) | $z_{0}^{139}+z_{0}z_{1}^{6}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{7}$ | Chain + Cycle | 4031 | 19488 | $(\mathbb{Z}_{29})^{6}$
(493,34,1841,1315,789) | $z_{1}z_{0}^{9}+z_{0}z_{1}^{117}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{4}$ | Cycle + Cycle | 4471 | 16848 | $(\mathbb{Z}_{17})^{5}$
(67,67,217,84,35) | $z_{0}^{7}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{3}+z_{3}z_{4}^{11}$ | BP + Cycle | 469 | 2376 | $(\mathbb{Z}_{67})^{6}$
(118,118,185,135,35) | $z_{0}^{5}+z_{1}^{5}+z_{4}z_{2}^{3}+z_{2}z_{3}^{3}+z_{3}z_{4}^{13}$ | BP + Cycle | 590 | 1872 | $(\mathbb{Z}_{118})^{4}$
(373,373,780,35,305) | $z_{0}^{5}+z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{31}+z_{3}z_{4}^{6}$ | BP + Cycle | 1865 | 5952 | $(\mathbb{Z}_{373})^{4}$
(113,226,715,377,39) | $z_{0}^{13}+z_{0}z_{1}^{6}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{28}$ | Chain + Cycle | 1469 | 7392 | $(\mathbb{Z}_{113})^{6}$
(253,253,545,40,175) | $z_{0}^{5}+z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{18}+z_{3}z_{4}^{7}$ | BP + Cycle | 1265 | 4032 | $(\mathbb{Z}_{253})^{4}$
(43,1333,1875,500,1625) | $z_{0}^{125}+z_{0}z_{1}^{4}+z_{4}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{3}$ | Chain + Cycle | 5375 | 15792 | $(\mathbb{Z}_{43})^{4}$
(43,1333,2375,1000,625) | $z_{0}^{125}+z_{0}z_{1}^{4}+z_{4}z_{2}^{2}+z_{2}z_{3}^{3}+z_{3}z_{4}^{7}$ | Chain + Cycle | 5375 | 15792 | $(\mathbb{Z}_{43})^{4}$
(73,73,95,45,80) | $z_{0}^{5}+z_{1}^{5}+z_{4}z_{2}^{3}+z_{2}z_{3}^{6}+z_{3}z_{4}^{4}$ | BP + Cycle | 365 | 1152 | $(\mathbb{Z}_{73})^{4}$
(185,740,1911,987,63) | $z_{0}^{21}+z_{0}z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{46}$ | Chain + Cycle | 3885 | 15640 | $(\mathbb{Z}_{185})^{5}$
(929,1858,2849,63,805) | $z_{0}^{7}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{58}+z_{3}z_{4}^{8}$ | Chain + Cycle | 6503 | 13920 | $(\mathbb{Z}_{929})^{3}$
(64,512,475,375,175) | $z_{0}^{25}+z_{0}z_{1}^{3}+z_{4}z_{2}^{3}+z_{2}z_{3}^{3}+z_{3}z_{4}^{7}$ | Chain + Cycle | 1600 | 3213 | $(\mathbb{Z}_{64})^{3}$
(253,253,600,95,65) | $z_{0}^{5}+z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{18}$ | BP + Cycle | 1265 | 4032 | $(\mathbb{Z}_{253})^{4}$
(127,381,793,286,65) | $z_{0}^{13}+z_{0}z_{1}^{4}+z_{4}z_{2}^{2}+z_{2}z_{3}^{3}+z_{3}z_{4}^{21}$ | Chain + Cycle | 1651 | 5040 | $(\mathbb{Z}_{127})^{4}$
(65,650,1581,867,153) | $z_{0}^{51}+z_{0}z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{16}$ | Chain + Cycle | 3315 | 13120 | $(\mathbb{Z}_{65})^{5}$
(231,66,481,185,259) | $z_{1}z_{0}^{5}+z_{0}z_{1}^{15}+z_{4}z_{2}^{2}+z_{2}z_{3}^{4}+z_{3}z_{4}^{4}$ | Cycle + Cycle | 1221 | 2400 | $(\mathbb{Z}_{33})^{3}$
(1003,68,3745,2675,1605) | $z_{1}z_{0}^{9}+z_{0}z_{1}^{119}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{4}$ | Cycle + Cycle | 9095 | 17136 | $(\mathbb{Z}_{17})^{3}$
(73,584,1435,779,123) | $z_{0}^{41}+z_{0}z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{18}$ | Chain + Cycle | 2993 | 11880 | $(\mathbb{Z}_{73})^{5}$
(481,962,1519,77,329) | $z_{0}^{7}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{24}+z_{3}z_{4}^{10}$ | Chain + Cycle | 3367 | 7200 | $(\mathbb{Z}_{481})^{3}$
(657,3942,4693,95,3097) | $z_{0}^{19}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{82}+z_{3}z_{4}^{4}$ | Chain + Cycle | 12483 | 25584 | $(\mathbb{Z}_{657})^{3}$
(2628,1971,4693,95,3097) | $z_{1}z_{0}^{4}+z_{0}z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{82}+z_{3}z_{4}^{4}$ | Cycle + Cycle | 12483 | 13120 | $(\mathbb{Z}_{657})^{2}$
(3773,98,8901,5031,1161) | $z_{1}z_{0}^{5}+z_{0}z_{1}^{155}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{12}$ | Cycle + Cycle | 18963 | 37200 | $(\mathbb{Z}_{49})^{3}$
(2069,2069,1413,102,555) | $z_{0}^{3}+z_{1}^{3}+z_{4}z_{2}^{4}+z_{2}z_{3}^{47}+z_{3}z_{4}^{11}$ | BP + Cycle | 6207 | 8272 | $(\mathbb{Z}_{2069})^{2}$
(929,1858,3199,413,105) | $z_{0}^{7}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{8}+z_{3}z_{4}^{58}$ | Chain + Cycle | 6503 | 13920 | $(\mathbb{Z}_{929})^{3}$
(3532,7064,5355,115,1595) | $z_{0}^{5}+z_{0}z_{1}^{2}+z_{4}z_{2}^{3}+z_{2}z_{3}^{107}+z_{3}z_{4}^{11}$ | Chain + Cycle | 17660 | 21186 | $(\mathbb{Z}_{3532})^{2}$
(1505,6020,3357,2547,117) | $z_{0}^{9}+z_{0}z_{1}^{2}+z_{4}z_{2}^{4}+z_{2}z_{3}^{4}+z_{3}z_{4}^{94}$ | Chain + Cycle | 13545 | 15040 | $(\mathbb{Z}_{1505})^{2}$
(136,119,889,635,381) | $z_{1}z_{0}^{15}+z_{0}z_{1}^{17}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{4}$ | Cycle + Cycle | 2159 | 4080 | $(\mathbb{Z}_{17})^{3}$
(1297,3891,2653,119,1120) | $z_{0}^{7}+z_{0}z_{1}^{2}+z_{4}z_{2}^{3}+z_{2}z_{3}^{54}+z_{3}z_{4}^{8}$ | Chain + Cycle | 9079 | 10368 | $(\mathbb{Z}_{1297})^{2}$
(6485,2594,9197,119,3655) | $z_{1}z_{0}^{3}+z_{0}z_{1}^{6}+z_{4}z_{2}^{2}+z_{2}z_{3}^{108}+z_{3}z_{4}^{6}$ | Cycle + Cycle | 22049 | 23328 | $(\mathbb{Z}_{1297})^{2}$
(1457,1457,1011,120,327) | $z_{0}^{3}+z_{1}^{3}+z_{4}z_{2}^{4}+z_{2}z_{3}^{28}+z_{3}z_{4}^{13}$ | BP + Cycle | 4371 | 5824 | $(\mathbb{Z}_{1457})^{2}$
(701,701,381,123,198) | $z_{0}^{3}+z_{1}^{3}+z_{4}z_{2}^{5}+z_{2}z_{3}^{14}+z_{3}z_{4}^{10}$ | BP + Cycle | 2103 | 2800 | $(\mathbb{Z}_{701})^{2}$
(2069,2069,1521,426,123) | $z_{0}^{3}+z_{1}^{3}+z_{4}z_{2}^{4}+z_{2}z_{3}^{11}+z_{3}z_{4}^{47}$ | BP + Cycle | 6207 | 8272 | $(\mathbb{Z}_{2069})^{2}$
(2149,921,3193,124,3131) | $z_{1}z_{0}^{4}+z_{0}z_{1}^{8}+z_{4}z_{2}^{2}+z_{2}z_{3}^{51}+z_{3}z_{4}^{3}$ | Cycle + Cycle | 9517 | 9792 | $(\mathbb{Z}_{307})^{2}$
(289,2312,2725,125,1775) | $z_{0}^{25}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{36}+z_{3}z_{4}^{4}$ | Chain + Cycle | 7225 | 14688 | $(\mathbb{Z}_{289})^{3}$
(129,3612,4165,425,2635) | $z_{0}^{85}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{16}+z_{3}z_{4}^{4}$ | Chain + Cycle | 10965 | 21888 | $(\mathbb{Z}_{129})^{3}$
(129,3612,5185,1445,595) | $z_{0}^{85}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{4}+z_{3}z_{4}^{16}$ | Chain + Cycle | 10965 | 21888 | $(\mathbb{Z}_{129})^{3}$
(4085,129,5745,1532,4979) | $z_{1}z_{0}^{4}+z_{0}z_{1}^{96}+z_{4}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{3}$ | Cycle + Cycle | 16469 | 16128 | $(\mathbb{Z}_{43})^{2}$
(4085,129,7277,3064,1915) | $z_{1}z_{0}^{4}+z_{0}z_{1}^{96}+z_{4}z_{2}^{2}+z_{2}z_{3}^{3}+z_{3}z_{4}^{7}$ | Cycle + Cycle | 16469 | 16128 | $(\mathbb{Z}_{43})^{2}$
(481,962,1617,175,133) | $z_{0}^{7}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{10}+z_{3}z_{4}^{24}$ | Chain + Cycle | 3367 | 7200 | $(\mathbb{Z}_{481})^{3}$
(657,3942,6175,1577,133) | $z_{0}^{19}+z_{0}z_{1}^{3}+z_{4}z_{2}^{2}+z_{2}z_{3}^{4}+z_{3}z_{4}^{82}$ | Chain + Cycle | 12483 | 25584 | $(\mathbb{Z}_{657})^{3}$
(2628,1971,6175,1577,133) | $z_{1}z_{0}^{4}+z_{0}z_{1}^{5}+z_{4}z_{2}^{2}+z_{2}z_{3}^{4}+z_{3}z_{4}^{82}$ | Cycle + Cycle | 12483 | 13120 | $(\mathbb{Z}_{657})^{2}$
(1945,477,1321,148,1871) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{8}+z_{1}z_{2}^{4}+z_{2}z_{3}^{30}+z_{3}z_{4}^{3}$ | Cycle | 5761 | 5760 | $\mathbb{Z}_{5761}$
(2387,1579,661,148,771) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{2}+z_{1}z_{2}^{6}+z_{2}z_{3}^{33}+z_{3}z_{4}^{7}$ | Cycle | 5545 | 5544 | $\mathbb{Z}_{5545}$
(9142,3097,1917,4129,149) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{3}+z_{1}z_{2}^{8}+z_{2}z_{3}^{4}+z_{3}z_{4}^{96}$ | Cycle | 18433 | 18432 | $\mathbb{Z}_{18433}$
(2323,1611,562,151,899) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{2}+z_{1}z_{2}^{7}+z_{2}z_{3}^{33}+z_{3}z_{4}^{6}$ | Cycle | 5545 | 5544 | $\mathbb{Z}_{5545}$
(3073,712,2211,151,1199) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{6}+z_{1}z_{2}^{3}+z_{2}z_{3}^{34}+z_{3}z_{4}^{6}$ | Cycle | 7345 | 7344 | $\mathbb{Z}_{7345}$
(1585,189,1105,292,1439) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{16}+z_{1}z_{2}^{4}+z_{2}z_{3}^{12}+z_{3}z_{4}^{3}$ | Cycle | 4609 | 4608 | $\mathbb{Z}_{4609}$
(18277,6172,10207,1899,239) | $z_{4}z_{0}^{2}+z_{0}z_{1}^{3}+z_{1}z_{2}^{3}+z_{2}z_{3}^{14}+z_{3}z_{4}^{146}$ | Cycle | 36793 | 36792 | $\mathbb{Z}_{36793}$

All the rational homology 7-spheres that we exhibit in Table 1 come from
polynomials that are of cycle type or that have that type of singularity as
part of its Thom-Sebastiani representation. The examples found in [7] also
have that particular feature. Actually, in [7], Lemma 3.3 provides examples of
cycle type singularities, while Lemma 3.10 provides examples of cycle+cycle
type singularities. Indeed, if we can write the vector
$\textbf{w}=(w_{0},w_{1},w_{2},w_{3},w_{4})=(m_{2}v_{0},m_{2}v_{1},m_{2}v_{2},m_{3}v_{3},m_{3}v_{4})$
where $\gcd(m_{2},m_{3})=1$ and $m_{2}m_{3}=d$, then Lemma 3.10 in [7] implies
that the number

$\displaystyle{n(\textbf{w})=\dfrac{m_{2}}{v_{3}v_{4}}-\dfrac{1}{v_{3}}-\dfrac{1}{v_{4}}}$

is a positive integer. This is equivalent to

$m_{2}=v_{3}+v_{4}(1+n(\textbf{w})v_{3}).$

Multiplying by $m_{3}$, we have

$d=w_{3}+w_{4}a_{4}$ (12)

where $a_{4}>1$ is integer. Analogously, we get
$m_{2}=v_{4}+v_{3}(1+n(\textbf{w})v_{4}).$ Multiplying by $m_{2}$, we obtain

$d=w_{4}+w_{3}a_{3}$ (13)

From (9) and (10) it follows that there exist a cycle block
$z_{4}z_{3}^{a_{3}}+z_{3}z_{4}^{a_{4}}$ that is a summand of some invertible
polynomial associated to w. We conjecture that all rational homology 7-spheres
arising as links of hypersurface singularities can be given by polynomials
that must contain a cycle singularity term in its representation as a Thom-
Sebastiani sum, at least when $d=\sum_{i}^{4}w_{i}+1.$

######  Remark 3.1.

In [5], from links $L_{f}^{\prime}$ of quasi-smooth weighted hypersurfaces
$f^{\prime}(z_{2}\ldots z_{n})$, the authors consider the quasi-smooth
weighted hypersurfaces

$f(z_{0},\ldots z_{n})=z_{0}^{2}+z_{1}^{2}+f^{\prime}$

in $\mathbb{C}^{n+1}.$ It is not difficult to show that the corresponding link
$L_{f}$ admits Sasakian structures with positive Ricci curvature.
Additionally, if the link $L_{f}^{\prime}$ is a rational homology sphere, from
Sebastiani-Thom theorem [25] it follows that the link $L_{f}$ is a rational
homology sphere. Thus, from each of the new 52 rational homology 7-spheres one
can produce a rational homology 11-sphere admitting positive Ricci curvature.

Following the terminology given in [7], a rational homology 7-spheres with the
same degree $d$, Milnor number $\mu$ and order of $H_{3}$ is a twin. We detect
ten twins in Table 1; except the couple given by the weights
$(2323,1611,562,151,899)$ and $(2387,1579,661,148,771),$ both of them of cycle
type, with $d=|H_{3}|=5545$ and $\mu=5544$, the rest of twins have weight
vectors with identical first two components $w_{0},w_{1}.$ As mentioned in
[7], it is tempting to conjecture that twins are homeomorphic or even
diffeomorphic links, however we are not able to determine this. In an upcoming
article, we study the behavior of twins trough an operation associated to the
Berglund-Hübsch transpose rule from BHK mirror symmetry [2].

###  3.3. Sasaki-Einstein 7-manifolds of the form $\\#k(S^{3}\times S^{4})$
from the Johnson-Kollár’s list and Cheltsov’s list

In [3] it is proven that a $2$-connected oriented 7-manifold $M$ that bounds a
parallelizable 8-manifold with $H_{3}(M,\mathbb{Z})$ torsion free is
completely determined up to diffeomorphism by the rank of
$H_{3}(M,\mathbb{Z}).$ Moreover $M$ is diffeomorphic to $\\#k\left(S^{3}\times
S^{4}\right)\\#\Sigma^{7}$ for some homotopy sphere $\Sigma^{7}\in bP_{8}$
that bounds a parallizable 8-manifold (one of the 28 possible smooth
structures on the oriented 7-sphere). Thus, if the link has no torsion it is
homeomorphic to $k\\#\left(S^{3}\times S^{4}\right)$ where $k$ is the rank of
the third homology group. In [5], it is showed that, being of Sasaki type, the
link has $k$ even.

Until now, the only cases where Sasaki-Einstein metrics were known to exist on
7-manifolds of the form $\\#k\left(S^{3}\times S^{4}\right)$ are
$\\#222\left(S^{3}\times S^{4}\right)$ and $\\#480\left(S^{3}\times
S^{4}\right)$. We found 124 new examples of 2-connected Sasaki-Einstein
7-manifolds of the form $\\#k(S^{3}\times S^{4}).$ Of that lot, 118 come from
the list given in [19], the other 6 examples are links of quasi-smooth Fano
3-folds coming from Reids list of 95 weighted codimension 1 K3 surfaces. We
can state the following result.

######  Theorem 3.3.

Sasaki-Einstein metrics exist on 7-manifolds of the form
$\\#k\left(S^{3}\times S^{4}\right)$ for 22 different values of $k$, where
$k=rank(H_{3}(M,\mathbb{Z}))$ is given in Table 2 and Table 3. $\square$

Notice that in Table 2, $k$ assumes every even number between 2 and 32 except
$k=22$ and $k=28,$ however the elements in Table 3 give sporadic values for
$k.$

Table 2: SE 7-manifolds of the form $\\#k(S^{3}\times S^{4})$ from the
Johnson-Kollár list

${\bf{w}}=(w_{0},w_{1},w_{2},w_{3},w_{4})$ | Polynomial | Type | $d$ | $\mu$ | $H_{3}(M,\mathbb{Z})$
---|---|---|---|---|---
(9,15,5,10,7) | $z_{0}^{5}+z_{1}^{3}+z_{1}z_{2}^{6}+z_{2}z_{3}^{4}+z_{3}z_{4}^{5}$ | BP + Chain | 45 | 1216 | $\mathbb{Z}^{32}$
(16,56,7,21,13) | $z_{0}^{7}+z_{1}^{2}+z_{1}z_{2}^{8}+z_{2}z_{3}^{5}+z_{3}z_{4}^{7}$ | BP + Chain | 112 | 2970 | $\mathbb{Z}^{30}$
(35,15,15,9,32) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{6}+z_{2}z_{3}^{10}+z_{3}z_{4}^{3}$ | BP + Chain | 105 | 1752 | $\mathbb{Z}^{24}$
(50,30,15,9,47) | $z_{0}^{3}+z_{1}^{5}+z_{1}z_{2}^{8}+z_{2}z_{3}^{15}+z_{3}z_{4}^{3}$ | BP + Chain | 150 | 2472 | $\mathbb{Z}^{24}$
(35,56,63,9,153) | $z_{0}^{9}+z_{0}z_{1}^{5}+z_{2}^{5}+z_{2}z_{3}^{28}+z_{3}z_{4}^{2}$ | Chain + Chain | 315 | 5328 | $\mathbb{Z}^{24}$
(141,9,138,95,41) | $z_{0}^{3}+z_{1}^{47}+z_{1}z_{2}^{3}+z_{2}z_{3}^{3}+z_{3}z_{4}^{8}$ | BP + Chain | 423 | 6112 | $\mathbb{Z}^{16}$
(48,10,115,25,43) | $z_{0}^{5}+z_{1}^{24}+z_{1}z_{2}^{2}+z_{2}z_{3}^{5}+z_{3}z_{4}^{5}$ | BP + Chain | 240 | 3940 | $\mathbb{Z}^{20}$
(49,35,105,10,47) | $z_{0}^{5}+z_{1}^{7}+z_{1}z_{2}^{2}+z_{2}z_{3}^{14}+z_{3}z_{4}^{5}$ | BP + Chain | 245 | 3168 | $\mathbb{Z}^{16}$
(714,476,119,11,109) | $z_{0}^{2}+z_{1}^{3}+z_{1}z_{2}^{8}+z_{2}z_{3}^{119}+z_{3}z_{4}^{13}$ | BP + Chain | 1428 | 34294 | $\mathbb{Z}^{26}$
(65,65,39,15,12) | $z_{0}^{3}+z_{1}^{3}+z_{2}^{5}+z_{3}^{13}+z_{3}z_{4}^{15}$ | BP + Chain | 195 | 2928 | $\mathbb{Z}^{24}$
(119,51,153,12,23) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{2}+z_{2}z_{3}^{17}+z_{3}z_{4}^{15}$ | BP + Chain | 357 | 6680 | $\mathbb{Z}^{20}$
(57,38,95,12,27) | $z_{0}^{4}+z_{1}^{6}+z_{1}z_{2}^{2}+z_{3}^{19}+z_{3}z_{4}^{8}$ | BP + Chain + Chain | 228 | 2814 | $\mathbb{Z}^{18}$
(81,45,12,131,137) | $z_{0}^{5}+z_{1}^{9}+z_{1}z_{2}^{30}+z_{2}z_{3}^{3}+z_{3}z_{4}^{2}$ | BP + Chain | 405 | 4288 | $\mathbb{Z}^{16}$
(236,12,87,207,167) | $z_{0}^{3}+z_{1}^{59}+z_{1}z_{2}^{8}+z_{2}z_{3}^{3}+z_{3}z_{4}^{3}$ | BP + Chain | 708 | 6492 | $\mathbb{Z}^{12}$
(16,14,49,21,13) | $z_{0}^{7}+z_{1}^{8}+z_{1}z_{2}^{2}+z_{2}z_{3}^{3}+z_{3}z_{4}^{7}$ | BP + Chain | 112 | 1782 | $\mathbb{Z}^{18}$
(35,21,14,13,23) | $z_{0}^{3}+z_{1}^{5}+z_{1}z_{2}^{6}+z_{2}z_{3}^{7}+z_{3}z_{4}^{4}$ | BP + Chain | 105 | 1312 | $\mathbb{Z}^{16}$
(495,135,736,107,13) | $z_{0}^{3}+z_{1}^{11}+z_{4}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{106}$ | BP + Cycle | 1485 | 29680 | $\mathbb{Z}^{20}$
(17,14,105,75,45) | $z_{0}^{15}+z_{0}z_{1}^{17}+z_{4}z_{2}^{2}+z_{2}z_{3}^{2}+z_{3}z_{4}^{4}$ | Chain + Cycle | 255 | 3856 | $\mathbb{Z}^{16}$
(133,38,19,14,63) | $z_{0}^{2}+z_{1}^{7}+z_{2}^{14}+z_{3}^{19}+z_{3}z_{4}^{4}$ | BP + Chain | 266 | 4524 | $\mathbb{Z}^{24}$
(119,21,14,153,51) | $z_{0}^{3}+z_{1}^{17}+z_{1}z_{2}^{24}+z_{4}z_{3}^{2}+z_{3}z_{4}^{4}$ | BP + Chain + Cycle | 357 | 6272 | $\mathbb{Z}^{32}$
(33,77,77,14,31) | $z_{0}^{7}+z_{1}^{3}+z_{1}z_{2}^{2}+z_{2}z_{3}^{11}+z_{3}z_{4}^{7}$ | BP + Chain | 231 | 2400 | $\mathbb{Z}^{12}$
(315,90,135,14,77) | $z_{0}^{2}+z_{1}^{7}+z_{1}z_{2}^{4}+z_{3}^{45}+z_{3}z_{4}^{8}$ | BP + Chain + Chain | 630 | 6952 | $\mathbb{Z}^{16}$
(15,20,85,17,119) | $z_{0}^{17}+z_{0}z_{1}^{12}+z_{2}^{3}+z_{2}z_{3}^{10}+z_{3}z_{4}^{2}$ | Chain + Chain | 255 | 6016 | $\mathbb{Z}^{32}$
(24,30,15,35,17) | $z_{0}^{5}+z_{1}^{4}+z_{1}z_{2}^{6}+z_{2}z_{3}^{3}+z_{3}z_{4}^{5}$ | BP + Chain | 120 | 1236 | $\mathbb{Z}^{12}$
(65,39,15,18,59) | $z_{0}^{3}+z_{1}^{5}+z_{2}^{13}+z_{2}z_{3}^{10}+z_{3}z_{4}^{3}$ | BP + Chain | 195 | 2176 | $\mathbb{Z}^{16}$
(51,15,24,77,89) | $z_{0}^{5}+z_{1}^{17}+z_{1}z_{2}^{10}+z_{2}z_{3}^{3}+z_{3}z_{4}^{2}$ | BP + Chain | 255 | 2656 | $\mathbb{Z}^{16}$
(143,26,39,15,207) | $z_{0}^{3}+z_{0}z_{1}^{11}+z_{2}^{11}+z_{2}z_{3}^{26}+z_{3}z_{4}^{2}$ | Chain + Chain | 429 | 9176 | $\mathbb{Z}^{32}$
(215,15,315,33,68) | $z_{0}^{3}+z_{1}^{43}+z_{1}z_{2}^{2}+z_{2}z_{3}^{10}+z_{3}z_{4}^{9}$ | BP + Chain | 645 | 13848 | $\mathbb{Z}^{24}$
(176,48,15,171,119) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{32}+z_{2}z_{3}^{3}+z_{3}z_{4}^{3}$ | BP + Chain | 528 | 4908 | $\mathbb{Z}^{12}$
(115,69,92,15,55) | $z_{0}^{3}+z_{1}^{5}+z_{1}z_{2}^{3}+z_{3}^{23}+z_{3}z_{4}^{6}$ | BP + Chain + Chain | 345 | 2552 | $\mathbb{Z}^{12}$
(117,65,104,285,15) | $z_{0}^{5}+z_{1}^{9}+z_{1}z_{2}^{5}+z_{4}z_{3}^{2}+z_{3}z_{4}^{20}$ | BP + Chain + Cycle | 585 | 5920 | $\mathbb{Z}^{16}$
(395,15,65,474,237) | $z_{0}^{3}+z_{1}^{79}+z_{1}z_{2}^{18}+z_{4}z_{3}^{2}+z_{3}z_{4}^{3}$ | BP + Chain + Cycle | 1185 | 16128 | $\mathbb{Z}^{24}$
(132,110,275,15,129) | $z_{0}^{5}+z_{1}^{6}+z_{1}z_{2}^{2}+z_{3}^{44}+z_{3}z_{4}^{5}$ | BP + Chain + Chain | 660 | 4956 | $\mathbb{Z}^{12}$
(228,570,15,125,203) | $z_{0}^{5}+z_{1}^{2}+z_{1}z_{2}^{38}+z_{2}z_{3}^{9}+z_{3}z_{4}^{5}$ | BP + Chain | 1140 | 11244 | $\mathbb{Z}^{12}$
(655,15,130,367,799) | $z_{0}^{3}+z_{1}^{131}+z_{1}z_{2}^{15}+z_{2}z_{3}^{5}+z_{3}z_{4}^{2}$ | BP + Chain | 1965 | 23320 | $\mathbb{Z}^{20}$
(740,222,999,15,245) | $z_{0}^{3}+z_{1}^{10}+z_{1}z_{2}^{2}+z_{3}^{148}+z_{3}z_{4}^{9}$ | BP + Chain + Chain | 2220 | 26070 | $\mathbb{Z}^{18}$
(200,16,127,39,19) | $z_{0}^{2}+z_{1}^{25}+z_{4}z_{2}^{3}+z_{2}z_{3}^{7}+z_{3}z_{4}^{19}$ | BP + Cycle | 400 | 9576 | $\mathbb{Z}^{24}$
(135,240,765,17,1139) | $z_{0}^{17}+z_{0}z_{1}^{9}+z_{2}^{3}+z_{2}z_{3}^{90}+z_{3}z_{4}^{2}$ | Chain + Chain | 2295 | 37264 | $\mathbb{Z}^{20}$
(77,33,33,18,71) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{6}+z_{2}z_{3}^{11}+z_{3}z_{4}^{3}$ | BP + Chain | 231 | 1920 | $\mathbb{Z}^{12}$
(4928,896,2921,19,1093) | $z_{0}^{2}+z_{1}^{11}+z_{4}z_{2}^{3}+z_{2}z_{3}^{365}+z_{3}z_{4}^{9}$ | BP + Cycle | 9856 | 98550 | $\mathbb{Z}^{10}$
(55,22,99,20,25) | $z_{0}^{4}+z_{1}^{10}+z_{1}z_{2}^{2}+z_{3}^{11}+z_{3}z_{4}^{8}$ | BP + Chain + Chain | 220 | 2574 | $\mathbb{Z}^{18}$
(2990,20,745,349,1877) | $z_{0}^{2}+z_{1}^{299}+z_{1}z_{2}^{8}+z_{2}z_{3}^{15}+z_{3}z_{4}^{3}$ | BP + Chain | 5980 | 73854 | $\mathbb{Z}^{14}$
(77,21,35,66,33) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{6}+z_{4}z_{3}^{3}+z_{3}z_{4}^{5}$ | BP + Chain + Cycle | 231 | 1680 | $\mathbb{Z}^{12}$
(203,21,294,45,47) | $z_{0}^{3}+z_{1}^{29}+z_{1}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{12}$ | BP + Chain | 609 | 8992 | $\mathbb{Z}^{16}$
(119,119,51,21,48) | $z_{0}^{3}+z_{1}^{3}+z_{2}^{7}+z_{3}^{17}+z_{3}z_{4}^{7}$ | BP + Chain | 357 | 2472 | $\mathbb{Z}^{12}$
(27,66,207,23,299) | $z_{0}^{23}+z_{0}z_{1}^{9}+z_{2}^{3}+z_{2}z_{3}^{18}+z_{3}z_{4}^{2}$ | Chain + Chain | 621 | 10360 | $\mathbb{Z}^{20}$
(172,473,1978,23,1311) | $z_{0}^{23}+z_{0}z_{1}^{8}+z_{2}^{2}+z_{2}z_{3}^{86}+z_{3}z_{4}^{3}$ | Chain + Chain | 3956 | 55890 | $\mathbb{Z}^{18}$
(6615,2835,9472,23,901) | $z_{0}^{3}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{451}+z_{3}z_{4}^{22}$ | BP + Cycle | 19845 | 238128 | $\mathbb{Z}^{12}$
(28,182,91,39,25) | $z_{0}^{13}+z_{1}^{2}+z_{1}z_{2}^{2}+z_{2}z_{3}^{7}+z_{3}z_{4}^{13}$ | BP + Chain | 364 | 4068 | $\mathbb{Z}^{12}$
(195,25,190,157,409) | $z_{0}^{5}+z_{1}^{39}+z_{1}z_{2}^{5}+z_{2}z_{3}^{5}+z_{3}z_{4}^{2}$ | BP + Chain | 975 | 4528 | $\mathbb{Z}^{8}$
(1045,418,190,25,413) | $z_{0}^{2}+z_{1}^{5}+z_{2}^{11}+z_{2}z_{3}^{76}+z_{3}z_{4}^{5}$ | BP + Chain | 2090 | 13416 | $\mathbb{Z}^{8}$
(9610,620,25,3839,5127) | $z_{0}^{2}+z_{1}^{31}+z_{1}z_{2}^{744}+z_{2}z_{3}^{5}+z_{3}z_{4}^{3}$ | BP + Chain | 19220 | 253674 | $\mathbb{Z}^{18}$
(4928,896,3277,731,25) | $z_{0}^{2}+z_{1}^{11}+z_{4}z_{2}^{3}+z_{2}z_{3}^{9}+z_{3}z_{4}^{365}$ | BP + Cycle | 9856 | 98550 | $\mathbb{Z}^{10}$
(170,102,51,27,161) | $z_{0}^{3}+z_{1}^{5}+z_{1}z_{2}^{8}+z_{2}z_{3}^{17}+z_{3}z_{4}^{3}$ | BP + Chain | 510 | 2792 | $\mathbb{Z}^{8}$
(92,414,207,27,89) | $z_{0}^{9}+z_{1}^{2}+z_{1}z_{2}^{2}+z_{2}z_{3}^{23}+z_{3}z_{4}^{9}$ | BP + Chain | 828 | 5912 | $\mathbb{Z}^{8}$
(1251,27,414,371,1691) | $z_{0}^{3}+z_{1}^{139}+z_{1}z_{2}^{9}+z_{2}z_{3}^{9}+z_{3}z_{4}^{2}$ | BP + Chain | 3753 | 24744 | $\mathbb{Z}^{12}$
(28,161,658,47,423) | $z_{0}^{47}+z_{0}z_{1}^{8}+z_{2}^{2}+z_{2}z_{3}^{14}+z_{3}z_{4}^{3}$ | Chain + Chain | 1316 | 18810 | $\mathbb{Z}^{18}$
(455,105,28,191,587) | $z_{0}^{3}+z_{1}^{13}+z_{1}z_{2}^{45}+z_{2}z_{3}^{7}+z_{3}z_{4}^{2}$ | BP + Chain | 1365 | 9336 | $\mathbb{Z}^{12}$
(854,28,105,229,493) | $z_{0}^{2}+z_{1}^{61}+z_{1}z_{2}^{16}+z_{2}z_{3}^{7}+z_{3}z_{4}^{3}$ | BP + Chain | 1708 | 14580 | $\mathbb{Z}^{12}$
(16646,4756,29,1147,10715) | $z_{0}^{2}+z_{1}^{7}+z_{1}z_{2}^{984}+z_{2}z_{3}^{29}+z_{3}z_{4}^{3}$ | BP + Chain | 33292 | 406386 | $\mathbb{Z}^{18}$
(84,35,77,30,195) | $z_{0}^{5}+z_{1}^{12}+z_{1}z_{2}^{5}+z_{3}^{14}+z_{3}z_{4}^{2}$ | BP + Chain + Chain | 420 | 2940 | $\mathbb{Z}^{12}$
(405,162,30,65,149) | $z_{0}^{2}+z_{1}^{5}+z_{2}^{27}+z_{2}z_{3}^{12}+z_{3}z_{4}^{5}$ | BP + Chain | 810 | 5288 | $\mathbb{Z}^{8}$
(1245,830,30,123,263) | $z_{0}^{2}+z_{1}^{3}+z_{2}^{83}+z_{2}z_{3}^{20}+z_{3}z_{4}^{9}$ | BP + Chain | 2490 | 26724 | $\mathbb{Z}^{12}$
(319,87,58,31,463) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{15}+z_{2}z_{3}^{29}+z_{3}z_{4}^{2}$ | BP + Chain | 957 | 9880 | $\mathbb{Z}^{20}$
(1323,567,1921,32,127) | $z_{0}^{3}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{64}+z_{3}z_{4}^{31}$ | BP + Cycle | 3969 | 47616 | $\mathbb{Z}^{12}$
(341,93,62,495,33) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{15}+z_{4}z_{3}^{2}+z_{3}z_{4}^{16}$ | BP + Chain + Cycle | 1023 | 9920 | $\mathbb{Z}^{20}$
(1173,782,102,33,257) | $z_{0}^{2}+z_{1}^{3}+z_{2}^{23}+z_{2}z_{3}^{68}+z_{3}z_{4}^{9}$ | BP + Chain | 2346 | 25068 | $\mathbb{Z}^{12}$
(440,330,33,117,401) | $z_{0}^{3}+z_{1}^{4}+z_{1}z_{2}^{30}+z_{2}z_{3}^{11}+z_{3}z_{4}^{3}$ | BP + Chain | 1320 | 5514 | $\mathbb{Z}^{6}$
(935,33,308,1275,255) | $z_{0}^{3}+z_{1}^{85}+z_{1}z_{2}^{9}+z_{4}z_{3}^{2}+z_{3}z_{4}^{6}$ | BP + Chain + Cycle | 2805 | 16344 | $\mathbb{Z}^{12}$
(1700,510,2295,33,563) | $z_{0}^{3}+z_{1}^{10}+z_{1}z_{2}^{2}+z_{2}z_{3}^{85}+z_{3}z_{4}^{9}$ | BP + Chain | 5100 | 27222 | $\mathbb{Z}^{6}$
(55,77,44,175,35) | $z_{0}^{7}+z_{1}^{5}+z_{1}z_{2}^{7}+z_{4}z_{3}^{2}+z_{3}z_{4}^{6}$ | BP + Chain + Cycle | 385 | 2232 | $\mathbb{Z}^{12}$
(189,35,182,405,135) | $z_{0}^{5}+z_{1}^{27}+z_{1}z_{2}^{5}+z_{4}z_{3}^{2}+z_{3}z_{4}^{4}$ | BP + Chain + Cycle | 945 | 3488 | $\mathbb{Z}^{8}$
(324,270,675,35,317) | $z_{0}^{5}+z_{1}^{6}+z_{1}z_{2}^{2}+z_{2}z_{3}^{27}+z_{3}z_{4}^{5}$ | BP + Chain | 1620 | 5212 | $\mathbb{Z}^{4}$
(6102,36,507,3899,1661) | $z_{0}^{2}+z_{1}^{339}+z_{1}z_{2}^{24}+z_{2}z_{3}^{3}+z_{3}z_{4}^{5}$ | BP + Chain | 12204 | 105430 | $\mathbb{Z}^{10}$
(345,45,110,37,499) | $z_{0}^{3}+z_{1}^{23}+z_{1}z_{2}^{9}+z_{2}z_{3}^{25}+z_{3}z_{4}^{2}$ | BP + Chain | 1035 | 10720 | $\mathbb{Z}^{12}$
(315,175,280,37,769) | $z_{0}^{5}+z_{1}^{9}+z_{1}z_{2}^{5}+z_{2}z_{3}^{35}+z_{3}z_{4}^{2}$ | BP + Chain | 1575 | 6448 | $\mathbb{Z}^{8}$
(27306,18204,37,1475,7591) | $z_{0}^{2}+z_{1}^{3}+z_{1}z_{2}^{984}+z_{2}z_{3}^{37}+z_{3}z_{4}^{7}$ | BP + Chain | 54612 | 658294 | $\mathbb{Z}^{14}$
(416,624,39,93,77) | $z_{0}^{3}+z_{1}^{2}+z_{1}z_{2}^{16}+z_{2}z_{3}^{13}+z_{3}z_{4}^{15}$ | BP + Chain | 1248 | 11710 | $\mathbb{Z}^{10}$
(6615,2835,9901,452,43) | $z_{0}^{3}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{22}+z_{3}z_{4}^{451}$ | BP + Cycle | 19845 | 238128 | $\mathbb{Z}^{12}$
(255,45,60,47,359) | $z_{0}^{3}+z_{1}^{17}+z_{1}z_{2}^{12}+z_{2}z_{3}^{15}+z_{3}z_{4}^{2}$ | BP + Chain | 765 | 6496 | $\mathbb{Z}^{16}$
(297,81,45,47,422) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{18}+z_{2}z_{3}^{18}+z_{3}z_{4}^{2}$ | BP + Chain | 891 | 7504 | $\mathbb{Z}^{12}$
(464,348,87,45,449) | $z_{0}^{3}+z_{1}^{4}+z_{1}z_{2}^{12}+z_{2}z_{3}^{29}+z_{3}z_{4}^{3}$ | BP + Chain | 1392 | 5658 | $\mathbb{Z}^{6}$
(387,215,344,945,45) | $z_{0}^{5}+z_{1}^{9}+z_{1}z_{2}^{5}+z_{4}z_{3}^{2}+z_{3}z_{4}^{22}$ | BP + Chain + Cycle | 1935 | 6512 | $\mathbb{Z}^{8}$
(6600,4400,275,47,1879) | $z_{0}^{2}+z_{1}^{3}+z_{1}z_{2}^{32}+z_{2}z_{3}^{275}+z_{3}z_{4}^{7}$ | BP + Chain | 13200 | 158494 | $\mathbb{Z}^{14}$
(494,76,57,49,313) | $z_{0}^{2}+z_{1}^{13}+z_{1}z_{2}^{16}+z_{2}z_{3}^{19}+z_{3}z_{4}^{3}$ | BP + Chain | 988 | 8100 | $\mathbb{Z}^{12}$
(608,114,855,51,197) | $z_{0}^{3}+z_{1}^{16}+z_{1}z_{2}^{2}+z_{2}z_{3}^{19}+z_{3}z_{4}^{9}$ | BP + Chain | 1824 | 9762 | $\mathbb{Z}^{6}$
(2996,856,107,55,1979) | $z_{0}^{2}+z_{1}^{7}+z_{1}z_{2}^{48}+z_{2}z_{3}^{107}+z_{3}z_{4}^{3}$ | BP + Chain | 5992 | 72234 | $\mathbb{Z}^{18}$
(34496,9856,22205,59,2377) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{793}+z_{3}z_{4}^{29}$ | BP + Cycle | 68992 | 413946 | $\mathbb{Z}^{6}$
(1323,567,1954,65,61) | $z_{0}^{3}+z_{1}^{7}+z_{4}z_{2}^{2}+z_{2}z_{3}^{31}+z_{3}z_{4}^{64}$ | BP + Cycle | 3969 | 47616 | $\mathbb{Z}^{12}$
(312,780,65,115,289) | $z_{0}^{5}+z_{1}^{2}+z_{1}z_{2}^{12}+z_{2}z_{3}^{13}+z_{3}z_{4}^{5}$ | BP + Chain | 1560 | 5084 | $\mathbb{Z}^{4}$
(8960,2560,5793,67,541) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{181}+z_{3}z_{4}^{33}$ | BP + Cycle | 17920 | 107514 | $\mathbb{Z}^{6}$
(4044,2696,1145,131,73) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{53}+z_{3}z_{4}^{109}$ | BP + Cycle | 8088 | 80878 | $\mathbb{Z}^{10}$
(2331,999,74,187,3403) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{81}+z_{2}z_{3}^{37}+z_{3}z_{4}^{2}$ | BP + Chain | 6993 | 43080 | $\mathbb{Z}^{12}$
(6272,1792,4069,75,337) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{113}+z_{3}z_{4}^{37}$ | BP + Cycle | 12544 | 75258 | $\mathbb{Z}^{6}$
(3002,76,741,277,1909) | $z_{0}^{2}+z_{1}^{79}+z_{1}z_{2}^{8}+z_{2}z_{3}^{19}+z_{3}z_{4}^{3}$ | BP + Chain | 6004 | 24570 | $\mathbb{Z}^{6}$
(3861,81,1278,1145,5219) | $z_{0}^{3}+z_{1}^{143}+z_{1}z_{2}^{9}+z_{2}z_{3}^{9}+z_{3}z_{4}^{2}$ | BP + Chain | 11583 | 25456 | $\mathbb{Z}^{4}$
(34496,9856,22969,1587,85) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{29}+z_{3}z_{4}^{793}$ | BP + Cycle | 68992 | 413946 | $\mathbb{Z}^{6}$
(5096,1456,3319,87,235) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{79}+z_{3}z_{4}^{43}$ | BP + Cycle | 10192 | 61146 | $\mathbb{Z}^{6}$
(8960,2560,5941,363,97) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{33}+z_{3}z_{4}^{181}$ | BP + Cycle | 17920 | 107514 | $\mathbb{Z}^{6}$
(4150,100,205,1619,2227) | $z_{0}^{2}+z_{1}^{83}+z_{1}z_{2}^{40}+z_{2}z_{3}^{5}+z_{3}z_{4}^{3}$ | BP + Chain | 8300 | 36438 | $\mathbb{Z}^{6}$
(3535,1515,202,103,5251) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{45}+z_{2}z_{3}^{101}+z_{3}z_{4}^{2}$ | BP + Chain | 10605 | 64248 | $\mathbb{Z}^{12}$
(2666,172,645,109,1741) | $z_{0}^{2}+z_{1}^{31}+z_{1}z_{2}^{8}+z_{2}z_{3}^{43}+z_{3}z_{4}^{3}$ | BP + Chain | 5332 | 21546 | $\mathbb{Z}^{6}$
(6272,1792,4145,227,109) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{37}+z_{3}z_{4}^{113}$ | BP + Cycle | 12544 | 75258 | $\mathbb{Z}^{6}$
(5096,1456,3355,159,127) | $z_{0}^{2}+z_{1}^{7}+z_{4}z_{2}^{3}+z_{2}z_{3}^{43}+z_{3}z_{4}^{79}$ | BP + Cycle | 10192 | 61146 | $\mathbb{Z}^{6}$
(10368,6912,1741,131,1585) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{11}+z_{2}z_{3}^{145}+z_{3}z_{4}^{13}$ | BP + Cycle | 20736 | 41470 | $\mathbb{Z}^{2}$
(10368,6912,1873,1451,133) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{11}+z_{2}z_{3}^{13}+z_{3}z_{4}^{145}$ | BP + Cycle | 20736 | 41470 | $\mathbb{Z}^{2}$
(18970,5420,813,137,12601) | $z_{0}^{2}+z_{1}^{7}+z_{1}z_{2}^{40}+z_{2}z_{3}^{271}+z_{3}z_{4}^{3}$ | BP + Chain | 37940 | 152034 | $\mathbb{Z}^{6}$
(6885,1215,2160,137,10259) | $z_{0}^{3}+z_{1}^{17}+z_{1}z_{2}^{9}+z_{2}z_{3}^{135}+z_{3}z_{4}^{2}$ | BP + Chain | 20655 | 41584 | $\mathbb{Z}^{4}$
(3240,2160,559,191,331) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{11}+z_{2}z_{3}^{31}+z_{3}z_{4}^{19}$ | BP + Cycle | 6480 | 12958 | $\mathbb{Z}^{2}$
(3240,2160,571,311,199) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{11}+z_{2}z_{3}^{19}+z_{3}z_{4}^{31}$ | BP + Cycle | 6480 | 12958 | $\mathbb{Z}^{2}$
(168480,112320,46837,223,9101) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{1301}+z_{3}z_{4}^{37}$ | BP + Cycle | 336960 | 673918 | $\mathbb{Z}^{2}$
(41472,27648,11561,247,2017) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{289}+z_{3}z_{4}^{41}$ | BP + Cycle | 82944 | 165886 | $\mathbb{Z}^{2}$
(168480,112320,48101,7807,253) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{37}+z_{3}z_{4}^{1301}$ | BP + Cycle | 336960 | 673918 | $\mathbb{Z}^{2}$
(41472,27648,11809,1735,281) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{41}+z_{3}z_{4}^{289}$ | BP + Cycle | 82944 | 165886 | $\mathbb{Z}^{2}$
(24840,16560,6947,283,1051) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{151}+z_{3}z_{4}^{47}$ | BP + Cycle | 49680 | 99358 | $\mathbb{Z}^{2}$
(24840,16560,7051,907,323) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{47}+z_{3}z_{4}^{151}$ | BP + Cycle | 49680 | 99358 | $\mathbb{Z}^{2}$
(18792,12528,5279,355,631) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{91}+z_{3}z_{4}^{59}$ | BP + Cycle | 37584 | 75166 | $\mathbb{Z}^{2}$
(18792,12528,5311,547,407) | $z_{0}^{2}+z_{1}^{3}+z_{4}z_{2}^{7}+z_{2}z_{3}^{59}+z_{3}z_{4}^{91}$ | BP + Cycle | 37584 | 75166 | $\mathbb{Z}^{2}$

Reid’s list of 95 codimension one K3 surfaces of the form
$Y_{d}\subset\mathbb{CP}(w_{1},w_{2},w_{3},w_{4})$ with
$\sum_{i=1}^{4}w_{i}=d$ can be used to generate $\mathbb{Q}$-Fano 3-folds
[18]. These threefolds are hypersurfaces $X_{d}$ of degree $d$ in weighted
projective spaces of the form $\mathbb{CP}(1,w_{1},w_{2},w_{3},w_{4})$ with
$\sum_{i=1}^{4}w_{i}=d$. In [12], Cheltsov studied these sort of
$\mathbb{Q}$-Fano 3-folds and proves that 91 elements admit Kähler-Einstein
orbifold metrics and thereby the links associated to them admit Sasaki-
Einstein metrics. The four that fail the test are numbers $1,2,4$, and $5$ in
the list given in [18]. From this list, we found that 88 of these links are
links of singularities of chain type, cycle type or an iterated Thom-
Sebastiani sum of chain type singularities and cycle type singularities. We
compute the third homology group for this lot and found 6 new examples of
Sasaki-Einstein 7-manifolds with no torsion on the third homology group. In
Table 3 we present these six elements and include the two elements found by
Boyer in [11], the ones with weights $(1,1,1,4,6)$ and $(1,1,6,14,21)$.

Table 3: SE 7-manifolds of the form $\\#k(S^{3}\times S^{4})$ from Fletcher’s
list

${\bf{w}}=(w_{0},w_{1},w_{2},w_{3},w_{4})$ | Polynomial | Type | $d$ | $\mu$ | $H_{3}(M,\mathbb{Z})$
---|---|---|---|---|---
(1,1,1,4,6) | $z_{0}^{12}+z_{1}^{12}+z_{2}^{12}+z_{3}^{3}+z_{4}^{2}$ | BP | 12 | 2662 | $\mathbb{Z}^{222}$
(7,3,1,10,1) | $z_{0}^{3}+z_{1}^{7}+z_{1}z_{2}^{18}+z_{2}z_{3}^{2}+z_{3}z_{4}^{11}$ | BP + Chain | 21 | 5280 | $\mathbb{Z}^{252}$
(14,4,1,9,1) | $z_{0}^{2}+z_{1}^{7}+z_{1}z_{2}^{24}+z_{2}z_{3}^{3}+z_{3}z_{4}^{19}$ | BP + Chain | 28 | 9234 | $\mathbb{Z}^{330}$
(11,3,5,14,1) | $z_{0}^{3}+z_{1}^{11}+z_{1}z_{2}^{6}+z_{2}z_{3}^{2}+z_{3}z_{4}^{19}$ | BP + Chain | 33 | 4864 | $\mathbb{Z}^{148}$
(18,12,1,5,1) | $z_{0}^{2}+z_{1}^{3}+z_{1}z_{2}^{24}+z_{2}z_{3}^{7}+z_{3}z_{4}^{31}$ | BP + Chain | 36 | 15190 | $\mathbb{Z}^{422}$
(1,1,6,14,21) | $z_{0}^{42}+z_{1}^{42}+z_{2}^{7}+z_{3}^{3}+z_{4}^{2}$ | BP | 42 | 20172 | $\mathbb{Z}^{480}$
(22,4,5,13,1) | $z_{0}^{2}+z_{1}^{11}+z_{1}z_{2}^{8}+z_{2}z_{3}^{3}+z_{3}z_{4}^{31}$ | BP + Chain | 44 | 7998 | $\mathbb{Z}^{182}$
(33,22,6,5,1) | $z_{0}^{2}+z_{1}^{3}+z_{2}^{11}+z_{2}z_{3}^{12}+z_{3}z_{4}^{61}$ | BP + Chain | 66 | 15860 | $\mathbb{Z}^{240}$

This list is not necessary exhaustive for Cheltsov’s list, since neither
numbers 52, 81 nor 86 in [18] admit descriptions in terms of the type of
singularities where Orlik’s conjecture is known to be valid. However, our
computer program suggests that these members do have torsion.

Also, it is interesting to notice that, the $\mathbb{Q}$-Fano 3-folds
considered by Cheltsov are $d$-fold branch covers for certain weighted
projective spaces branched over orbifold $K3$ surfaces of degree $d$. It
follows from a well-known result on links of branched covers (see Proposition
2.1 in [14]), that the 91 Sasaki-Einstein links associated to the 3-folds of
Cheltsov’s list can be realized as $d$-fold branched covers of $S^{7}$
branched along the submanifolds $\\#k(S^{2}\times S^{3}).$ Actually, we extend
Theorem 4.6 in [11].

######  Theorem 3.4.

There exist Sasaki-Einstein metrics on the 7-manifolds $M^{7}$ which can be
realized as $d$-fold branched covers of $S^{7}$ branched along the
submanifolds $\\#k(S^{2}\times S^{3})$ where $k$ ranges from 3 to 21, except
$k=17.$ The $\mathbb{Q}$-Fano 3-folds, the homology of $M^{7}$ and $k$ are
given in Table I in Appendix, $(b)$.

###### Proof.

This is just a consequence of Theorem A in [13] (also see Appendix B in [5]),
and the fact that for each $k\not=17$ ranging from 3 to 21 there is an element
in Cheltsov’s list that can be given as an hypersurface singularity that is of
one of the types admissible for Orlik’s conjecture. ∎

In Appendix (c) we also include Tables II and III which list the third
homology group of links of hypersurface singularities which are neither
rational homology 7-spheres nor homeomorphic to connected sums of $S^{3}\times
S^{4}$. Table II consists of 79 links coming from Cheltsov’s list and Table
III consists of 1503 links extracted from the Johnson and Kollár list.

##  4\. appendix

  * (a)

The Johnson-Kollár list of hypersurfaces in weighted projective 4-space
admitting Kähler-Einstein orbifold metrics is available at
https://web.math.princeton.edu/ jmjohnso/delpezzo/
KEandTiger.txt. This list includes the weight vectors followed by data on
whether or not it is known if the orbifold is Kähler-Einstein.

  * (b)

Table I, cited in Theorem 3.2 is given here
https://github.com/Jcuadrosvalle/TABLES

  * (c)

Table II and Table III, where we exhibit the third homology group of links of
hypersurface singularities that are neither rational homology 7-spheres nor
homeomorphic to connected sums of $S^{3}\times S^{4},$ are available at
https://github.com/Jcuadrosvalle/TABLES

  * (d)

Four codes in Matlab are available at https://github.com/Jcuadrosvalle/Codes-
in-Matlab. Codes1, 2 and 3 determine whether or not the singularities are of
chain type, cycle type or an iterated Thom-Sebastiani sum of chain type and
cycle type singularities, Code4 computes the third homology groups of the
corresponding links.

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