A mirror deformation of Markov numbers

TL;DR

This paper introduces mirror Markov numbers as a novel $q$-deformation of classical Markov numbers, characterized by a new deformed squared Markov equation, with geometric, algebraic, and combinatorial insights and applications.

Contribution

It defines mirror Markov numbers via a new deformed equation, characterizes them, and explores their geometric and algebraic properties, including mutation rules and connections to classical equations.

Findings

Mirror Markov numbers are a new $q$-deformation of classical Markov numbers.

A mutation rule called mirror mutation generates all mirror Markov numbers.

Deformed equations relate to Fibonacci, Pell, and generalized Markov equations.

Abstract

We introduce a deformed squared Markov equation given by $X^2 + Y^2 + Z^2 + (q+q^{-1})(XY+YZ+XZ) = 3(1 + q + q^{-1})XYZ$. Symmetric solutions of this new equation present a remarkable factorization property which allows us to talk about their square roots. These square roots give a natural $q$-deformation of the Markov numbers that has not previously occurred in the literature. We call them mirror Markov numbers. We prove a characterization of mirror Markov numbers and discover a mutation rule, mirror mutation, to generate them all. We also prove a geometric realization of the corresponding mirror mutation on a once-punctured sphere with three orbifold points. Our mirror deformation leads to deformations of Fibonacci and Pell branches for which we give precise formulas. Furthermore, the deformed squared Markov equation specializes to many other very well known generalized Markov equations. We also obtain the super Markov numbers from a specialization of the deformed squared Markov numbers, which we use to prove a conjecture of Musiker.