Words for generalized Markov numbers

Abstract

We construct a word-theoretic framework for generalized Markov numbers, that is, positive integers appearing in positive integer solutions of the generalized Markov equation $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$. 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 $\overline{\omega}_t=xyz\omega_t^{-1}$ is related to the generalized Cohn matrices.