  The purpose of this note is to connect two maps related to certain graphs
embedded in the disc. The first is Postnikov's boundary measurement map, which
combines partition functions of matchings in the graph into a map from an
algebraic torus to an open positroid variety in a Grassmannian. The second is a
rational map from the open positroid variety to an algebraic torus, given by
certain Pl\"ucker coordinates which are expected to be a cluster in a cluster
structure.
  This paper clarifies the relationship between these two maps, which has been
ambiguous since they were introduced by Postnikov in 2001. The missing
ingredient supplied by this paper is a twist automorphism of the open positroid
variety, which takes the target of the boundary measurement map to the domain
of the (conjectural) cluster. Among other applications, this provides an
inverse to the boundary measurement map, as well as Laurent formulas for twists
of Pl\"ucker coordinates.
