  For a given marked surface $(S,M)$ and a fixed tagged triangulation $T$ of
$(S,M)$, we show that each tagged triangulation $T'$ of $(S,M)$ is uniquely
determined by the intersection numbers of tagged arcs of $T$ and tagged arcs of
$T'$. 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.
