Arithmetic on $q$-deformed rational numbers

Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}{s}>0$, they constructed coprime polynomials ${\mathcal{R}}_{\frac{r}{s}}(q),{\mathcal{S}}_{\frac{r}{s}}(q)\in \mathbb{Z}[q]$ with ${\mathcal{R}}_{\frac{r}{s}}(1)=r,{\mathcal{S}}_{\frac{r}{s}}(1)=s$. Their theory has a rich background and many applications. By definition, if $r\equiv r^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}s)$, then ${\mathcal{S}}_{\frac{r}{s}}(q)={\mathcal{S}}_{\frac{r^{\mathrm{\prime }}}{s}}(q)$. We show that $r{r}^{\mathrm{\prime }}\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}s)$ implies ${\mathcal{S}}_{\frac{r}{s}}(q)={\mathcal{S}}_{\frac{r^{\mathrm{\prime }}}{s}}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r\equiv ̸r^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}s)$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if ${\mathcal{S}}_{\frac{r}{s}}(\zeta )=0$ for $\zeta =(-1+\sqrt{-3})\mathrm{/}2$ (resp. $\zeta =i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.