On the monotonicity of the generalized Markov numbers

TL;DR

This paper investigates the behavior of generalized Markov numbers along lines of specific slopes, utilizing the Markov distance and Ptolemy inequality to establish their monotonicity properties.

Contribution

It provides a complete characterization of the monotonicity of generalized Markov numbers using geometric inequalities and distance measures.

Findings

Monotonicity of generalized Markov numbers determined for all slopes.

Application of Markov distance and Ptolemy inequality in this context.

Complete classification of monotonicity behavior along lines.

Abstract

Using the Markov distance and Ptolemy inequality introduced by Lee-Li-Rabideau-Schiffler \cite{LLRS}, we completely determine the monotonicity of the generalized Markov numbers along the lines of a given slope.