$F$-matrices of cluster algebras from triangulated surfaces

For a given marked surface $(S,M)$ and a fixed tagged triangulation $T$ of $(S,M)$, we show that each tagged triangulation $T^{\mathrm{\prime }}$ of $(S,M)$ is uniquely determined by the intersection numbers of tagged arcs of $T$ and tagged arcs of $T^{\mathrm{\prime }}$. As consequence, each cluster in the cluster algebra $\mathcal{A}(T)$ is uniquely determined by its $F$-matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda.