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Construction of Lie algebra weight system kernel via Vogel algebra

Dmitry Khudoteplov, Elena Lanina, Alexey Sleptsov

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 $Λ$ 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.

Construction of Lie algebra weight system kernel via Vogel algebra

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 algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of 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.

Paper Structure

This paper contains 24 sections, 20 equations, 28 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Vassiliev invariants and related notions
  3. Vassiliev invariants
  4. Chord diagrams
  5. Weight systems
  6. Vassiliev-Kontsevich theorem
  7. Jacobi diagrams
  8. AS and IHX relations
  9. Open Jacobi diagrams
  10. Lie algebra weight systems
  11. Vogel algebra
  12. Characters on $\Lambda$
  13. Construction of weight system kernel
  14. Method
  15. Example for $\mathfrak{sl}_n$ weight system
  16. ...and 9 more sections

Figures (28)

  • Figure 1: Vassiliev skein relation.
  • Figure 2: Knots $K$, $K'$ and $K^{\bullet}$.
  • Figure 3: Special knot and its chord diagram.
  • Figure 4: Four terms knot relation.
  • Figure 5: 4-term relation on functions coming from Vassiliev invariants.
  • ...and 23 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Vassiliev invariants vassiliev1990cohomology
  • Definition 2.5
  • Definition 2.6: Space of chord diagrams $\mathcal{A}$ bar1995vassiliev
  • Definition 2.7: Weight systems $\mathcal{W}$ bar1995vassiliev
  • Definition 2.8: Space of Jacobi diagrams $\mathcal{C}$ bar1995vassilievchmutov2012introduction
  • Definition 2.9: Open Jacobi diagrams $\mathcal{B}$ bar1995vassiliev
  • Definition 3.1: Algebra $\Lambda$ vogel2011algebraic
  • ...and 3 more