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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.
Forward citations
Cited by 2 Pith papers
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Mutation-preserving generalized cluster algebras and Laurent mutation invariants
A new stability condition for generalized cluster algebras is classified, a Markov-type Diophantine equation is solved with explicit orbit counts, and the Chen-Li conjecture on rank-3 Laurent mutation invariants is proved.
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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.
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