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A generalization of Markov Numbers

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Abstract

We explore a generalization of the Markov numbers that is motivated by a specific generalized cluster algebra arising from an orbifold, in the sense of Chekhov and Shapiro. We give an explicit algorithm for computing these generalized Markov numbers and exhibit several patterns analogous to those that appear within the ordinary Markov numbers. Along the way, we present formulas related to continued fractions and snake graphs.

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Acknowledgements

The first author would like to thank Gregg Musiker for the original idea to work on these generalizations of Markov numbers and Yasuaki Gyoda for discussions about the project.

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Authors and Affiliations

  1. Department of Mathematics, Aarhus University, 8000, Aarhus, Denmark

    Esther Banaian

  2. 970 Evans Hall, Berkeley, CA, 94720, USA

    Archan Sen

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  1. Esther Banaian
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  2. Archan Sen
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Correspondence to Esther Banaian.

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Banaian, E., Sen, A. A generalization of Markov Numbers. Ramanujan J 63, 1021–1055 (2024). https://doi.org/10.1007/s11139-023-00801-6

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