Saturation of Markov Polynomials

TL;DR

This paper proves a conjecture about Markov polynomials, extending understanding of solutions to the Markov equation through combinatorial methods involving Markov snake graphs.

Contribution

It provides a constructive proof of a conjecture on Markov polynomials, strengthening the link between Markov equations and combinatorics.

Findings

Proved a conjecture about Markov polynomials.

Connected Markov solutions to combinatorial structures.

Enhanced understanding of Markov equation solutions.

Abstract

Solutions to the Markov equation appear in many mathematical contexts. We aim to build on the understanding of them by proving a recent conjecture about Markov polynomials; solutions to a generalised version of the Markov equation. The proof we provide is a constructive argument based on the Markov snake graph, a combinatorial object related to Markov numbers, deepening the connection between the Markov equation and combinatorics.