Topological model for $q$-deformed rational numbers and categorification

Abstract

Let ${\mathbf{D}}_3$ be a bigraded 3-decorated disk with an arc system $\mathbf{A}$. We associate a bigraded simple closed arc ${\hat{\eta }}_{r/s}$ on ${\mathbf{D}}_3$ to any rational number $r/s\in \overline{\mathbb{Q}}=\mathbb{Q}\cup \{\infty \}$. We show that the right (respectively, left) $q$-deformed rational numbers associated to $r/s$, in the sense of Morier-Genoud–Ovsienko (respectively, Bapat–Becker–Licata) can be naturally calculated by the $\mathfrak{q}$-intersection between ${\hat{\eta }}_{r/s}$ and $\mathbf{A}$ (respectively, dual arc system ${\mathbf{A}}^{\ast }$). The Jones polynomials of rational knots can be also given by such intersections. Moreover, the categorification of ${\hat{\eta }}_{r/s}$ is given by the spherical object $X_{r/s}$ in the Calabi–Yau-$\mathbb{X}$ category of Ginzburg dga of type $A_2$. Reducing to the CY-2 case, we recover result of Bapat–Becker–Licata with a slight improvement.