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