Orderings on k-Markov numbers

Abstract The k -Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $$x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$$ x 2 + y 2 + z 2 + k ( x y + x z + y z ) = ( 3 + 3 k ) x y z . When $$k =0$$ 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.