potentials, each of which must be of one of the three so-called *atomic types*:

$$\begin{aligned}
W_{\text{Fermat}} &:= x^a, \\
W_{\text{loop}} &:= x_1^{a_1}x_2 + x_2^{a_2}x_3 + \ldots +x_{m-1}^{a_{m-1}}x_m + x_m^{a_m}x_1, \text{ and } \\
W_{\text{chain}} &:=x_1^{a_1}x_2 + x_2^{a_2}x_3 + \ldots x_{m-1}^{a_{m-1}}x_m + x_m^{a_m}.
\end{aligned}$$

Using Condition (1), we define the matrix $B = dA^{-1}$, where $d$ is a positive integer so that all the entries of $B$ are integers (note that $d$ is not necessarily the smallest such $d$). Take $e := (1, \ldots, 1)^T \in {\mathbb{R}}^n$ and $$q := Be\text{, i.e., }q_i = \sum_j b_{ij}.$$ Then the polynomial $F_A$ defines a zero locus $X_A = Z(F_A) \subseteq W{\mathbb{P}}^n(q_0, \ldots, q_n)$. Indeed, with these weights, the polynomial $F_A$ is quasihomogeneous: each monomial in $F_A$ has degree $\sum_{j=0}^n a_{ij}q_j = d$, as $Aq = ABe = de$. Condition (2) above is used to ensure that each integer $q_i$ is positive.

Assume further that $\sum_i q_i = d$ is the degree of the polynomial, which implies that the hypersurface $X_A$ is a Calabi-Yau variety. Define $\text{Sing}(V)$ to be the singular locus of any variety $V$, we say the hypersurface $X_A$ is quasi-smooth if $\text{Sing}(X_A) \subseteq \text{Sing}(W{\mathbb{P}}^n(q_0,\ldots, q_n)) \cap X_A$. Condition (3) above implies 