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arXiv · 2507.19080

On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges

Sam Evans·Perrine Jouteur·Sophie Morier-Genoud·Valentin Ovsienko
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Abstract

We consider a natural $q$-deformation of the classical Markov numbers. This $q$-deformation is closely related to $q$-deformed rational numbers recently introduced by two of us. Both notions, those of $q$-rationals and $q$-Markov numbers, are based on invariance with respect to the action of the modular group $mathrm{PSL}(2,\mathbb{Z})$. We prove that every Markov number has a unique $q$-deformation, which is a monic unimodal palindromic Laurent polynomial with positive integer coefficients. The $q$-Markov numbers can be calculated in terms of the traces of $q$-deformed Cohn matrices, and we show that $q$-Markov numbers are independent of the choice of such matrices. We construct a combinatorial model counting perfect matchings of snake graphs with weighted edges.

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Sam Evans, Perrine Jouteur, Sophie Morier-Genoud, Valentin Ovsienko. 2025-07-25. On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges. https://arxiv.org/abs/2507.19080

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