  We introduce a set of highly symmetric families that each calculate the
determinant of a given $\{\pm 1\}$-matrix by using the oriented hypergraphic
Laplacian and the incidence-based notion of cycle-covers. Any non-edge-monic
family of cycle-covers is shown to vanish in every determinant, while any one
of the $n!$ remaining edge-monic families is equivalent to determining the
absolute value of the determinant. Hadamard's maximum determinant problem is
shown to be equivalent to optimizing the number of locally signed graphic
circles of a given sign in any one of these families or across all of these
families. Various symmetries regarding orthogonality, equivalence to
$\{0,+1\}$-matrices, and their relation to these edge-monic families are shown
to be different fundamental circles related by theta-subgraphs.
