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arXiv · 2509.04614

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

Daniel Pérez Melesio·José Simental
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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 $\mathbb{F} \not\cong \mathbb{F}_2$.

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Daniel Pérez Melesio, José Simental. 2026-06-01. Cluster tori over $\mathbb{F}_2$, hexagonal moves on triangulations, and minimal coverings of cluster manifolds. https://arxiv.org/abs/2509.04614

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