Construction of Lie algebra weight system kernel via Vogel algebra

TL;DR

This paper introduces a new method for constructing the kernel of Lie algebra weight systems using Vogel's algebra, with explicit examples and implications for knot theory and topological quantum field theory.

Contribution

It develops a novel approach to identify the kernel of Lie algebra weight systems employing Vogel's algebra, providing explicit low-order examples and exploring theoretical consequences.

Findings

Explicit description of kernel elements for al sl_N at low orders

Implications for knot invariants and 3D Chern-Simons theory

Framework applicable to other Lie algebras and topological invariants

Abstract

We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $\Lambda$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight systems for detection of correlators in the 3D Chern-Simons topological field theory and for distinguishing of knots by the corresponding quantum knot invariants.