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<p>We consider a natural <i>q</i>-deformation of the classical Markov numbers. This <i>q</i>-deformation is closely related to <i>q</i>-deformed rational numbers recently introduced by two of us. Both notions, those of <i>q</i>-rationals and <i>q</i>-Markov numbers, are based on invariance with respect to the action of the modular group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{PSL}(2,{\mathbb {Z}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PSL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that every Markov number has a unique <i>q</i>-deformation, which is a monic unimodal palindromic Laurent polynomial with positive integer coefficients. The <i>q</i>-Markov numbers can be calculated in terms of the traces of <i>q</i>-deformed Cohn matrices, and we show that <i>q</i>-Markov numbers are independent of the choice of such matrices. We construct a combinatorial model counting perfect matchings of snake graphs with weighted edges.</p>

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On q-Deformed Markov Numbers: Cohn Matrices and Perfect Matchings with Weighted Edges

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Sam Evans,
  • Perrine Jouteur,
  • Sophie Morier-Genoud,
  • Valentin Ovsienko

摘要

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 \(\textrm{PSL}(2,{\mathbb {Z}})\) PSL ( 2 , 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.