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

TL;DR

This paper introduces a unique $q$-deformation of classical Markov numbers, represented as Laurent polynomials, and connects them to $q$-deformed Cohn matrices and combinatorial models involving perfect matchings.

Contribution

It establishes the existence and uniqueness of $q$-deformed Markov numbers as Laurent polynomials and links them to $q$-deformed Cohn matrices and combinatorial models.

Findings

Each Markov number has a unique $q$-deformation as a Laurent polynomial.

$q$-Markov numbers are computed via traces of $q$-deformed Cohn matrices.

A combinatorial model using perfect matchings of snake graphs is constructed.

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.