Genuses of cluster quivers of finite mutation type

Fang Li, Jichun Liu and Yichao Yang

Vol. 269 (2014), No. 1, 133–148

DOI: 10.2140/pjm.2014.269.133

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 nonnegative 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 $n$.

Keywords

cluster quiver, finite mutation type, genus, triangulation of surface

Mathematical Subject Classification 2010

Primary: 05C10, 05E10, 05E15, 13F60

Milestones

Received: 6 November 2012

Revised: 1 July 2013

Accepted: 25 August 2013

Published: 15 July 2014