Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers

The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\boldsymbol{w}$ be a fork-preserving mutation sequence. We show that every $c$-vector of $Q$ obtained from $\boldsymbol{w}$ is a solution to a quadratic equation of the form $$\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,$$ where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. The same proof techniques implies that when $Q$ is a rank 3 mutation-cyclic quiver, every $c$-vector of $Q$ is a solution to a quadratic equation of the same form.
Comment: 29 pages; Extended abstract of paper appeared at FPSAC 2024, published in S\'eminaire Lotharingien de Combinatoire Volume 91B