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

TL;DR

This paper provides a topological interpretation of coloured Jones and Alexander polynomials as intersection pairings in covering spaces, unifying their conceptual framework through explicit homology classes and intersection theory.

Contribution

It introduces a unified topological approach to coloured Jones and Alexander polynomials via intersection pairings in covering spaces, revealing their relation as specializations of the same pairing.

Findings

Coloured Jones and Alexander polynomials are specializations of a common intersection pairing.

The approach explains Bigelow's noodles and forks picture from a quantum perspective.

The coloured Alexander polynomial is a graded intersection pairing in a specific covering space.

Abstract

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 Lagrangian submanifolds. The main result proves that the $N^{th}$ coloured Jones polynomial and $N^{th}$ 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. 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 $N^{th}$ coloured Alexander polynomial is a graded intersection pairing in a $ \mathbb Z \oplus \mathbb Z_N$-covering of the configuration space in the punctured disc.