Graded mutation in cluster categories coming from hereditary categories with a tilting object

Tubular Cluster Algebras II: Exponential growth

“…Note that the endomorphism ring of a cluster tilting object in C is in general not determined by its quiver. For example, this occurs for the tubular cluster category of weight type (2, 2, 2, 2), see [5,Expl. 6.12].…”