A topological model for the coloured Jones polynomials

TL;DR

This paper introduces a homological topological model for Coloured Jones Polynomials, representing them as graded intersection pairings in a covering space, linking quantum and homological braid group representations.

Contribution

It presents a novel homological framework for Coloured Jones Polynomials using the Lawrence representation and braid group homology.

Findings

Homological model for Coloured Jones Polynomials established

Representation as graded intersection pairings in covering spaces

Connection between quantum and homological braid group representations

Abstract

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.