A $q$-analog of the Markoff injectivity conjecture holds

TL;DR

This paper proves that a $q$-analog of the Markoff injectivity conjecture is valid for Christoffel words at a specific root of unity, advancing understanding of the conjecture's structure.

Contribution

The authors establish injectivity of the $q$-analog on Christoffel words at a specific root of unity, providing new insights into the Markoff injectivity conjecture.

Findings

Injectivity holds for the $q$-analog at $q = ext{exp}(i heta)$ with $ heta = rac{ ext{pi}}{3}$.

Other roots of unity give partial information on the original conjecture.

Injectivity does not hold for some pairs of arbitrary words.

Abstract

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a long-standing open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A $q$-analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this $q$-analog. The proof is based on the evaluation at $q = \exp(i\pi/3)$. Other roots of unity provide some information on the original problem, which corresponds to the case $q=1$. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.