Words for generalized Markov numbers

We construct a word-theoretic framework for generalized Markov numbers, that is, positive integers appearing in positive integer solutions of the generalized Markov equation $x^2+y^2+z^2+k_{1}yz+k_{2}zx+k_{3}xy=(3+k_1+k_2+k_3)xyz$. For each positive rational slope $t$, we define a word ${\omega }_t$ by a recursive rule on a binary tree and realize it geometrically by a line segment of slope $t$. Matrix evaluation of ${\omega }_t$ gives a Markov–monodromy matrix encoding the generalized Markov number at $t$. We also show that ${\omega }_t$ recovers the classical Cohn word by a local substitution rule, and that the completed word ${\bar{\omega }}_t=xyz{\omega }_t^{-1}$ is related to the generalized Cohn matrices.