Cluster tori over ${\mathbb{F}}_2$, hexagonal moves on triangulations, and minimal coverings of cluster manifolds

Abstract

We study cluster algebras over ${\mathbb{F}}_2$. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type $A$, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose corresponding cluster tori cover the cluster manifold over any field $\mathbb{F}$, but there are also sections giving seeds whose cluster tori do not cover the cluster manifold over any field $\mathbb{F}\ncong {\mathbb{F}}_2$.

21 pages, 15 figures. Comments welcome!; v2 incorporates referee's comments and fixes many typos. To appear in Elec. J. Comb