Rationality Results for the Configuration Space Integral of Knots

TL;DR

This paper investigates the configuration space integral of knots within the perturbative Chern-Simons framework, introducing a new compactification approach and establishing rationality results with insights into denominators.

Contribution

It presents a new compactification method and proves rationality results for the configuration space integral of knots, enhancing understanding of its algebraic structure.

Findings

Established rationality of the configuration space integral

Provided new bounds on denominators in the rationality results

Reformulated the theory using a novel degree theory approach

Abstract

The perturbative Chern-Simons theory for knots in Euclidean space is a linear combination of integrals on configuration spaces. This has been successively studied by Bott and Taubes, Altschuler and Freidel, and Yang. We study it again in terms of degree theory, with a new choice of compactification. This paper is self-contained and proves some old and new results, especially a rationality result with some information on the denominators.