Orderings of k-Markov Numbers

Abstract: The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $$x² + y² + z² + k(xy + xz + yz) = (3+3k)xyz$$. 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.