Numerical knot invariants of finite type from Chern-Simons perturbation theory

TL;DR

This paper derives new finite type knot invariants from Chern-Simons perturbation theory, providing integral representations and computational results up to type six for prime knots, advancing the understanding of quantum invariants.

Contribution

It introduces a novel method to obtain numerical finite type knot invariants from Chern-Simons gauge theory perturbation series, expanding the toolkit for knot invariant computation.

Findings

New finite type knot invariants derived from Chern-Simons theory.

Computed invariants up to type six for all prime knots with up to six crossings.

Invariants can potentially be normalized to be integer-valued.

Abstract

Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants), and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to six crossings. Our results suggest that these knot invariants can be normalized in such a way that they are integer-valued.