Cyclotomic factors and irreducibility of the denominators of $q$-deformed rational numbers

For $d\ge 2$ the $q$-deformed modular group specialized at a primitive $d$-th root of unity is the triangle group of type $(2,3,d)$. Using this we determine the fractions $r\mathrm{/}s$ for which the $d$-th cyclotomic polynomial divides the denominator $S_{r\mathrm{/}s}(q)$ of the $q$-deformed rational number $[r\mathrm{/}s{]}_q$. They form the orbit of $\mathrm{\infty }$ under the normal closure of the translation $z\mapsto z+d$ in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences $s\equiv 0$ and $r\equiv \pm 1$ modulo $d$ characterize the divisibility exactly for $d\le 5$. For $a\in \{2,3,4,6\}$ and $n>5a^2$ prime to $a$ we show that $S_{a\mathrm{/}n}(q)$ is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime $p$ and every $r$ prime to $p$ with $r\equiv \pm a$ or $ar\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$.