A q -analog of the Markoff injectivity conjecture holds

Source 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 ( i π / 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.