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

The (right) $q$-deformed rational numbers was introduced by Morier-Genoud and Ovsienko, and its left variant, whose numerators and denominators are essentially the normalized Jones polynomials of rational links, by Bapat, Becker and Licata. These notions are based on continued fractions and the $q$-deformed modular group $\operatorname{PSL}_q(2,\mathbb{Z})$-actions. In this paper, we introduce the \textit{$q$-transpose} for matrices in $\operatorname{PSL}_q(2,\mathbb{Z})$ to refine the basic perspective of the theory. For example, we present a new proof and a refinement of a theorem of Leclere and Morier-Genoud stating that the trace of $A \in \operatorname{PSL}(2,\mathbb{Z})$ is always palindromic and sign coherent. We also show arithmetic/combinatorial results on left $q$-deformed rationals (e.g., the criterion for their palindromicity). Finally, we discuss the connection to the conjecture of Kantarc{\i} O\u{g}uz on circular fence posets.