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.
Super Markov Numbers and Signed Double Dimer Covers
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abstract
We provide a superalgebraic analogue of Markov numbers, which are defined as the Grassmann integer solutions to the equation $x^2 + y^2 + z^2 + (xy + yz + xz)\epsilon = 3(1 + \epsilon)xyz$, as well as applications to the Decorated Super Teichm\"uller spaces associated to the once-punctured torus and certain annuli. We conclude with further directions for study.
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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.