Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces

Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by immersed Lagrangian submanifolds. The main result proves that the Nth coloured Jones polynomial and Nth coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. This model can be evaluated at roots of unity. The first corollary explains Bigelow’s picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the Nth coloured Alexander polynomial is a graded intersection pairing in a Z ⊕ Z2N -covering of the configuration space in the punctured disc. This paper comes with a sequel article where, based on this result, we construct another topological model with homology classes given by explicit embedded Lagrangians, which are more suitable for computations. As a corollary of these two papers, we provide two intersection models (an immersed one and an embedded one) each of which leads to the Jones polynomial and Alexander polynomial by suitable specialisations.