Genuses of cluster quivers of finite mutation type

TL;DR

This paper investigates the genuses of cluster quivers of finite mutation type, establishing bounds and existence results related to surface genus and quiver genus, with implications for understanding their topological properties.

Contribution

It proves that in 11 exceptional cases, the genus distribution is limited to 0 or 1, and shows the genus of a surface bounds the genus of associated cluster quivers, also establishing existence of quivers with any genus.

Findings

In 11 exceptional cases, genuses are 0 or 1.

Surface genus bounds cluster quiver genus.

Existence of cluster quivers with any genus n.

Abstract

In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the $11$ exceptional cases, the distribution of genuses is $0$ or $1$. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It is verified that the genus of an oriented surface is an upper bound for the genuses of cluster quivers from this surface. Furthermore, for any non-negative integer $n$ and a closed oriented surface of genus $n$, we show that there always exist a set of punctures and a triangulation of this surface such that the corresponding cluster quiver from this triangulation is exactly of genus $n$.