# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.04056