  It has recently been demonstrated that Feynman integrals relevant to a wide
range of perturbative quantum field theories involve periods of Calabi-Yaus of
arbitrarily large dimension. While the number of Calabi-Yau manifolds of
dimension three or higher is considerable (if not infinite), those relevant to
most known examples come from a very simple class: degree-$2k$ hypersurfaces in
$k$-dimensional weighted projective space $\mathbb{WP}^{1,\ldots,1,k}$. In this
work, we describe some of the basic properties of these spaces and identify
additional examples of Feynman integrals that give rise to hypersurfaces of
this type. Details of these examples at three and four loops are included as
ancillary files to this work.
