A Generalization of Markov Numbers

TL;DR

This paper generalizes Markov numbers through a new cluster algebra framework from orbifolds, providing algorithms, patterns, and formulas related to continued fractions and snake graphs.

Contribution

It introduces a novel generalization of Markov numbers based on orbifold cluster algebras, with explicit computation methods and pattern analysis.

Findings

Explicit algorithm for generalized Markov numbers

Identification of patterns similar to classical Markov numbers

Formulas connecting continued fractions and snake graphs

Abstract

We explore a generalization of the Markov numbers that is motivated by a specific generalized cluster algebra arising from an orbifold, in the sense of Chekhov and Shapiro. We give an explicit algorithm for computing these generalized Markov numbers and exhibit several patterns analogous to those that appear within the ordinary Markov numbers. Along the way, we present formulas related to continued fractions and snake graphs.