Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

TL;DR

This paper develops a method to compute and analyze group-theoretical structures in Chern-Simons theory, leading to new insights into knot invariants, polynomial relations, and their algebraic properties.

Contribution

It introduces a novel approach for calculating group factors and explores their implications for knot invariants and polynomial relations in Chern-Simons theory.

Findings

Computed Vassiliev invariants using the new method

Established recursive relations for colored Jones polynomials of torus knots

Generalized the one-hook scaling property for colored Alexander polynomials

Abstract

We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values. First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots $T[2,2k+1]$. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank $N$ and the representation $R$.