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

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}^{*} ). The Jones polynomials of rational knots can be also given by such intersections. Moreover, the categorification of \widehat{\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.

• Revista Matemática Iberoamericana
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