2026/07/24 by Perrine Jouteur, Olga Paris-Romaskevich, Alexander Thomas
#math.QA #math.CO #math.GT
We prove unicity of q-rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual q-integers: the original q-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.