# $F$-matrices of cluster algebras from triangulated surfaces

## TL;DR

This paper demonstrates that in cluster algebras from triangulated surfaces, each cluster can be uniquely identified by its $F$-matrix, which is derived from intersection numbers of tagged arcs, providing a new invariant.

## Contribution

It introduces the $F$-matrix as a novel numerical invariant that uniquely determines clusters in cluster algebras from surfaces, based on intersection numbers.

## Key findings

- Each tagged triangulation is uniquely determined by intersection numbers.
- Every cluster in the algebra is uniquely identified by its $F$-matrix.
- The $F$-matrix serves as a new invariant for classifying clusters.

## Abstract

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.