The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square root of the partition function for such graphs in terms of a monopole-dimer model on a new kind of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises Wu’s assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights.

中文翻译：

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单极二聚体模型的垂直度

最近引入的单极二聚体模型是二聚体模型的一个完全可解的有符号推广。我们表明，单点二聚体模型在定点自由对合下不变图上的分配函数是一个理想的平方。我们用这种新型图上的单极二聚体模型对这种图的分配函数的平方根进行组合解释，这种图具有两种类型的边（我们称为双子叶植物）。后者的分区函数可以写成行列式，这是一个复杂的邻接矩阵。这种表述概括了吴将网格图的虚构方向分配给平面双子叶植物。作为应用程序，我们为具有正权重的非平面双子叶植物系列计算分区函数。