Polynomials for Torus Links from Chern-Simons Gauge Theories

TL;DR

This paper derives explicit polynomial invariants for torus links using Chern-Simons gauge theory, connecting topological invariants with conformal field theory and providing new computational methods.

Contribution

It introduces a novel approach to compute torus link polynomials via Chern-Simons theory and relates them to conformal field theory structures.

Findings

Explicit Jones polynomial expressions for $SU(2)$ links.

Polynomials for $N$-state vertex models ($N>2$).

Factorization of link invariants in minimal models.

Abstract

Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the $SU(2)$ gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the $N$-state ($N>2$) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two $SU(2)$ polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory.