Christoffel Words and Markoff Triples: An Algebraic Approach

TL;DR

This paper explores the algebraic relationship between Christoffel words and Markoff triples by introducing Markoff modules and establishing a bijection, providing new insights into the Markoff numbers' uniqueness conjecture.

Contribution

It introduces Markoff modules and constructs a bijective map linking Markoff module triples to Markoff triples, offering an algebraic perspective on the conjecture.

Findings

Markoff modules are combinatorially similar to Christoffel words

A bijective map between Markoff module triples and Markoff triples is established

Provides an algebraic interpretation of the Markoff numbers' uniqueness conjecture

Abstract

We introduce a family of modules, called Markoff modules, generated by a cluster-mutation-like iterative process. We show that these modules are combinatorially similar to Christoffel words. Furthermore, we construct a bijective map between the set of Markoff module triples and the set of proper Markoff triples. This allows us to interpret the uniqueness conjecture for Markoff numbers within an algebraic framework.