  Planar bicolored (plabic) graphs are combinatorial objects introduced by
Postnikov to give parameterizations of the positroid cells of the totally
nonnegative Grassmannian $\text{Gr}^{\geq 0}(n,k)$. Any two plabic graphs for
the same positroid cell can be related by a sequence of certain moves. The flip
graph has plabic graphs as vertices and has edges connecting the plabic graphs
which are related by a single move. A recent result of Galashin shows that
plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic
zonotope $Z(n,3)$. Taking this perspective, we show that the fundamental group
of the flip graph is generated by cycles of length 4, 5, and 10, and use this
result to prove a related conjecture of Dylan Thurston about triple crossing
diagrams. We also apply our result to make progress on an instance of the
generalized Baues problem.
