Inverse problem in cylindrical electrical networks

In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical $R$-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow, and of de Verdi\`ere, Gitler, and Vertigan for circular planar electrical networks. We show that our conjectural solution holds for certain "purely cylindrical" networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson.