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 Rₑₒ (q), Sₑₒ (q) Zq with Rₑₒ (1) =r, Sₑₒ (1) =s. Their theory has a rich background and many applications. By definition, if r r' s, then Sₑₒ (q) = Sₑ'ₒ (q). We show that rr'=-1 s implies Sₑₒ (q) = Sₑ'ₒ (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 Sₑₒ () =0 for = (-1+-3) /2 (resp. =i). We give applications to the representation theory of quivers of type A and the Jones polynomials of rational links.