Chern-Simons Theory, Colored-Oriented Braids and Link invariants

TL;DR

This paper develops a comprehensive method for solving SU(2) Chern-Simons theory on S^3 using coloured-oriented braids, resulting in new link invariants including but extending beyond Jones polynomials, with explicit calculations for various knots and links.

Contribution

It introduces a novel approach connecting coloured-oriented braids with Chern-Simons theory to generate new link invariants, expanding the toolkit for topological quantum field theory.

Findings

Derived explicit link invariants including Jones polynomials.

Calculated invariants for knots up to eight crossings.

Extended invariants to two-component multicoloured links.

Abstract

A method to obtain explicit and complete topological solution of SU(2) Chern-Simons theory on $S^3$ is developed. To this effect the necessary aspects of the theory of coloured-oriented braids and duality properties of conformal blocks for the correlators of $SU(2)_k$ Wess-Zumino conformal field theory are presented. A large class of representations of the generators of the groupoid of coloured-oriented braids are obtained. These provide a whole lot of new link invariants of which Jones polynomials are the simplest examples. These new invariants are explicitly calculated as illustrations for knots upto eight crossings and two-component multicoloured links upto seven crossings.