Boundary measurements of positive networks on a cylinder of maximal rank 2 and 3

Boundary measurement matrices associated to networks on a plane correspond to certain totally nonnegative Grassmannians as shown previously by A. Postnikov. In this paper, we look to generalize this result by categorizing the boundary measurements associated to networks on a cylinder of maximal rank 2 and 3. In particular, we show that the maximal rank 3 matrices associated to networks on a cylinder are precisely the matrices in which every odd-dimensional minor is nonnegative.