Coincidences of $q$-analogues of Markoff numbers outside the known families: an answer to a question of Labbé, Lapointe and Steiner
Labbé, Lapointe and Steiner proved that $w\mapsto\mu_q(w)_{12}$ is injective on Christoffel words, gave two families of identities showing that it is not injective on $\{\mathtt{0},\mathtt{1}\}^*$, and asked whether every coincidence $\mu_q(x)_{12}=\mu_q(y)_{12}$ comes from these two families. We show that it does not. The smallest counterexamples have length $24$; by an exhaustive computation, among words of length at most $32$ there are exactly $50$ classes of coincidences not explained by the two families, $10$ of length $24$ and $40$ of length $30$. Each of them contains an instance of…
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