Ternary and quaternary positroids

A positroid is an ordered matroid realizable by a real matrix with all nonnegative maximal minors. Postnikov gave a map from ordered matroids to Grassmann necklaces, for which there is a unique positroid in each fiber of the map. Here, we give forbidden minor characterizations of ternary and quaternary positroids. We show that a positroid is ternary if and only if it is near-regular, and that all ternary positroids are formed by direct sums and $2$-sums of binary positroids and positroid ordered whirls. We prove that a positroid is quaternary if and only if it is $U^2_6, U^4_6,$ and $P_6$-free. Under the map from ordered matroids to Grassmann necklaces, we fully characterize the fibers of ternary positroids, referred to as their positroid envelope classes; in particular, the envelope class of a positroid ordered whirl of rank-$r$ contains exactly four matroids.