Every cluster variable on the surface can be expressed as cluster expansion of the labeled seed. M-path is one way to calculate the cluster expansion of given arc using the multiplication of 2 × 2 matrices. I will extend this idea of M-path to find the cluster expansion of the tagged arcs. We will define a loop graph, which can be understood as a combination of a generalized snake graph and a band graph, and then define M-path on this graph.
