Transposes in the $q$-deformed modular group and their applications to $q$-deformed rational numbers

Abstract: The (right) \textit{q}-deformed rational numbers was originally introduced by Morier-Genoud and Ovsienko, and its left variant by Bapat, Becker and Licata. These notions are based on continued fractions and the \textit{q}-deformed modular group actions. For any matrix in this group, Leclere and Morier-Genoud showed that its trace is a palindromic polynomial whose coefficients are all non-negative (up to $\pm q^{N}$ times ). In this paper, we define the \textit{$q$-transpose} for matrices in this group. Using this, we give new proofs of the above result and some others. We also give arithmetic properties on the left \textit{q}-rational numbers. Finally, we show that the conjecture of Kantarc{\i} O\u{g}uz on circular fence posets implies a conjecture on the normalized Jones polynomials of rational links.