Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces
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The theory of quantum invariants for links started with the Jones polynomial and after that Reshetikhin and Turaev developed an algebraic construction which starts with a quantum group and gives an invariant. 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 non-semisimple invariants, called coloured Alexander polynomials. On the topological side, Lawrence introduced a sequence of homological 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 U_q(sl(2))-quantum invariants. More specifically, we define certain homology classes in versions of the Lawrence representation, 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 of Lagrangian intersections in configuration spaces (which is defined over 3 variables).