The $q$-analog of the Markoff injectivity conjecture holds over the language of a balanced sequence

The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $mu_q$ of $mu$ such that $mu_{qto1}(w)=mu(w)$ is the Markoff number associated to the Christoffel word $w$. We show that for every $q>0$, the map ${mathtt{0},mathtt{1}}^*tomathbb{Z}[q]$ defined by $wmapstomu_q(w)_{12}$ is injective over the language $mathcal{L}(s)$ of a balanced sequence $sin{mathtt{0},mathtt{1}}^mathbb{Z}$. The proof is based on new equivalent definitions of balanced sequences.