$U_q(sl(2))-$quantum invariants unified via intersections of embedded Lagrangians

In this paper we prove a unified model for $U_q(sl(2))$ quantum invariants through intersections of embedded Lagrangians in configuration spaces. More specifically, we construct a {\em state sum of Lagrangian intersections in the configuration space in the punctured disc}, which is a polynomial in three variables. It {\em recovers the coloured Jones polynomial and the coloured Alexander polynomial} through specialisations of coefficients. This formula works for oriented links coloured with the same representation of the quantum group and can be evaluated at roots of unity. As a corollary, the Jones and Alexander polynomials come both as {\em specialisations of an intersection pairing between embedded Lagrangians} in configuration spaces, which is suitable for computations. In particular, we obtain the {\em first intersection model for the Jones polynomial} from intersections between submanifolds which are given by {\em arcs and circles} in the punctured disc.