Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of link invariants. In this context, the quantum group $U_q(sl(2))$ leads to the sequence of coloured Jones polynomials, which contains the original Jones polynomial. Dually, the quantum group at roots of unity gives the sequence of coloured Alexander polynomials. On the topological side, Lawrence defined representations of braid groups on the homology of coverings of configurations spaces. Then, Bigelow and Lawrence gave a topological model for the original Jones polynomial.

 We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the $N$th coloured Jones and $N$th coloured Alexander invariants come as different specialisations of a state sum (defined over $3$ variables) of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.