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 $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{-3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.