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 $\hat{\hat{I}}{\cdot }_{\frac{r}{s}}$ 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 $\hat{\hat{I}}{\cdot }_{\frac{r}{s}}$ and $\mathbf{A}$ (resp. dual arc system ${\mathbf{A}}^{\ast }$). The Jones polynomials of rational knots can be also given by such intersections. Moreover, the categorification of $\hat{\hat{I}}{\cdot }_{\frac{r}{s}}$ 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.

28 pages, 17 figures