  We study the dimer model for a planar bipartite graph N embedded in a disk,
with boundary vertices on the boundary of the disk. Counting dimer
configurations with specified boundary conditions gives a point in the totally
nonnegative Grassmannian. Considering pairing probabilities for the
double-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's
Temperley-Lieb immanants. The same problem for the (probably novel)
triple-dimer model gives rise to the combinatorics of Kuperberg's webs and
Grassmann analogues of Pylyavskyy's web immanants. This draws a connection
between the square move of plabic graphs (or urban renewal of planar bipartite
graphs), and Kuperberg's square reduction of webs. Our results also suggest
that canonical-like bases might be applied to the dimer model.
  We furthermore show that these functions on the Grassmannian are compatible
with restriction to positroid varieties. Namely, our construction gives bases
for the degree two and degree three components of the homogeneous coordinate
ring of a positroid variety that are compatible with the cyclic group action.
