Markov数的镜像变形
A mirror deformation of Markov numbers
AI总结：
本文提出了一种Markov数的$q$-变形，通过镜像突变规则生成所有镜像Markov数，并给出了其几何实现和相关变形的公式。
摘要：
我们引入了一个变形的平方Markov方程，其形式为$X^2 + Y^2 + Z^2 + (q+q^{-1})(XY+YZ+XZ) = 3(1 + q + q^{-1})XYZ$。这种新方程的对称解具有显著的因式分解性质，允许我们讨论其平方根。这些平方根给出了一种自然的$q$-变形的Markov数，这在文献中尚属首次出现。我们证明了镜像Markov数的特征，并发现了一种突变规则，即镜像突变，用于生成所有镜像Markov数。我们还证明了对应的镜像突变在带有三个orbifold点的一次 punctured 球面上的几何实现。我们的镜像变形导致了Fibonacci和Pell分支的变形，我们给出了精确的公式。此外，变形的平方Markov方程可以退化为许多其他非常著名的广义Markov方程。我们还从变形的平方Markov数的特例中获得了超Markov数，这被用来证明Musiker的一个猜想。
英文摘要：
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.
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