A q-analog of the Markoff injectivity conjecture holds

Abstract

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a longstanding 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(iπ/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.