Construction of Lie algebra weight system kernel via Vogel algebra

We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel’s $\mathrm{\Lambda }$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of ${\mathfrak{s}\mathfrak{l}}_N$ 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.