Determinants of Simple Theta Curves and Symmetric Graphs

A theta curve is a spatial embedding of the $\theta$-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff's Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.