Colored Jones polynomials and abelianized Lefschetz numbers

TL;DR

This paper reveals a connection between colored Jones polynomials of braids and abelianized Lefschetz numbers, providing a new topological interpretation involving configuration spaces and duality pairings.

Contribution

It introduces a novel interpretation of colored Jones polynomials as sums of abelianized Lefschetz numbers related to braid actions on configuration spaces.

Findings

Colored Jones polynomials can be expressed as weighted sums of abelianized Lefschetz numbers.

The sum over configuration points relates to Poincare--Lefschetz duality intersection pairing.

Provides a topological framework linking knot invariants and configuration space dynamics.

Abstract

We show that the family of colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with the action of the braid on configuration spaces. The sum is over the number of configuration points. Then we interpret this sum in terms of Poincare--Lefschetz duality intersection pairing between homology classes.