Bott-Cattaneo-Rossi invariants for long knots in asymptotic homology $\mathbb R^3$

TL;DR

This paper relates Bott-Cattaneo-Rossi invariants to Alexander polynomials of long knots in punctured rational homology 3-spheres, expressing these invariants via configuration space integrals and connecting them to Chern-Simons theory.

Contribution

It provides a new integral formula for Alexander polynomials of long knots using Bott-Cattaneo-Rossi invariants and links these invariants to perturbative Chern-Simons theory.

Findings

Expressed Alexander polynomial in terms of configuration space integrals.

Connected Bott-Cattaneo-Rossi invariants with the Alexander polynomial.

Related invariants to the perturbative expansion of Chern-Simons theory.

Abstract

In this article, we express the Alexander polynomial of null-homologous long knots in punctured rational homology $3$-spheres in terms of integrals over configuration spaces. To get such an expression, we use a previously established formula, which gives generalized Bott-Cattaneo-Rossi invariants in terms of the Alexander polynomial and vice versa, and we relate these Bott-Cattaneo-Rossi invariants to the perturbative expansion of Chern-Simons theory.