Orderings of k-Markov Numbers

Abstract

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2+y^2+z^2+k(xy+xz+yz)=(3+3k)xyz$. When $k=0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures.

27 pages, many figures, comments are very welcome! v2: Changed numbering system, updated several proofs (most prominently, the proof of what is now Prop 43), corrected small typos