A mirror deformation of Markov numbers

TL;DR

The paper introduces a $q$-deformation of the Markov equation via the deformed squared Markov equation $X^2+Y^2+Z^2+(q+q^{-1})(XY+YZ+XZ)=3(1+q+q^{-1})XYZ$, yielding symmetric Laurent polynomial solutions organized into a mutation tree. It then defines mirror Markov numbers as square roots of these symmetric solutions, with a mutation theory (mirror mutation) and a geometric model on a sphere with one puncture and three orbifold points that clarifies the mutation dynamics. The framework unifies several known generalizations (including $k$-generalized and super Markov numbers) through specializations $q+q^{-1}=k$ and $q+q^{-1}=oldsymbol{\varepsilon}$, and it provides concrete Fibonacci- and Pell-type branches with explicit mutation formulas and positivity results, supported by continued-fraction representations. The work also establishes a geometric-cluster-algebra perspective for mirror mutation and proposes numerous open questions, including positivity conjectures and uniqueness on mirror branches, with implications for generalized Markov theories and super Teichmüller theory.

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.

Figures (16)

• Figure 1: First levels of Markov tree $\mathbb{T}_{0}$
• Figure 2: Markov quiver
• Figure 3: Initial steps of the deformed squared Markov tree
• Figure 4: Super Ptolemy transformation, where $-$ indicates opposite orientation.
• Figure 5: Equivalence relation, where $-$ indicates the edge with opposite orientation

Theorems & Definitions (48)

• Remark 2.1
• Theorem 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Remark 2.5
• proof : Proof of Proposition \ref{['prop:allsolutionsontree']}
• Definition 3.1: Proposition 7.12 of GMS
• Lemma 3.2
• proof