A topological model for the coloured Alexander invariants

TL;DR

This paper introduces a topological model for coloured Alexander invariants, representing them as graded intersection pairings in a covering space, thus linking quantum invariants to topological structures.

Contribution

It provides a novel topological framework for understanding coloured Alexander invariants using intersection pairings in configuration space coverings.

Findings

Coloured Alexander polynomials are realized through topological intersection pairings.

The model connects quantum invariants with classical topological concepts.

The approach recovers the original Alexander polynomial as the first term.

Abstract

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for this invariants, showing that they can be obtained as graded intersection pairings between homology classes in a covering of the configuration space in the punctured disc.