Arithmetic on q-deformed rational numbers

Recently, Morier-Genoud and Ovsienko introduced a q-analog of rational numbers. More precisely, for an irreducible fraction rs>0, they constructed coprime polynomials R_rs(q), S_rs(q) ∈ Z[q] with R_rs(1)=r, S_rs(1)=s. Their theory has a rich background and many applications. By definition, if r ≡ r' s, then S_rs(q)= S_r's(q). We show that rr'=-1 s implies S_rs(q)= S_r's(q), and it is conjectured that the converse holds if s is prime (and r ≡ r' s). We also show that s is a multiple of 3 (resp. 4) if and only if S_rs(ζ)=0 for ζ=(-1+√-3)/2 (resp. ζ=i). We give applications to the representation theory of quivers of type A and the Jones polynomials of rational links.