On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916863608356864 |
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| author | Evans, Sam Jouteur, Perrine Morier-Genoud, Sophie Ovsienko, Valentin |
| author_facet | Evans, Sam Jouteur, Perrine Morier-Genoud, Sophie Ovsienko, Valentin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges Evans, Sam Jouteur, Perrine Morier-Genoud, Sophie Ovsienko, Valentin Combinatorics 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. |
| title | On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.19080 |