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.
Categories: math.RT math.CO