To formulate two-dimensional Yang–Mills theory with adjoint matter fields in the large-Nlimit as classical mechanics, we derive a Poisson algebra for the color-invariant observables involving adjoint matter fields. We showed rigorously in Ref. [15] that different quantum orderings of the observables produce essentially the same Poisson algebra. Here we explain, in a less precise but more pedagogical manner, the crucial topological graphical observations underlying the formal proof. ©1999 American Institute of Physics.