Two new families of cluster Cohn and Markov-monodromy matrices for generalized Markov cluster algebras are introduced, fully classified, and made explicit via weighted fence posets whose order ideals expand cluster variables.
A cluster theory approach from mutation invariants to Diophantine equations
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abstract
In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive integer points of the Markov mutation invariant and its variant. As an application, several classes of Diophantine equations with cluster algebraic structures are exhibited.
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Cluster algebraic interpretation of generalized Markov numbers and their matrixizations
Two new families of cluster Cohn and Markov-monodromy matrices for generalized Markov cluster algebras are introduced, fully classified, and made explicit via weighted fence posets whose order ideals expand cluster variables.