Contents

• Contents
• 1 Introduction
  • 1.1 Finite type invariants and the fundamental theorem
  • 1.2 Generators, relations and syzygies
  • 1.3 Parenthesized tangles
  • 1.4 Plan of the paper
  • 1.5 Acknowledgement
• 2 Knotted trivalent graphs
  • 2.1 Knotted trivalent graphs and some moves between them
  • 2.2 The set of knotted trivalent graphs is finitely generated
  • 2.3 Chord diagrams on trivalent skeleta
  • 2.4 Symmetries, the pentagon and the hexagon relations
  • 2.5 Vertex renormalizations and uniqueness
  • 2.6 Is KTG finitely presented?
  • 2.7 The relation with $6j$-symbols and Turaev-Viro invariants
  • 2.8 The relation with perturbative Chern-Simons theory
  • 2.9 Some dreams regarding Witten's Asymptotics Conjecture and the Kashaev-Murakami-Murakami Volume Conjecture
• 3 Planar Algebras
  • 3.1 A quick introduction to planar algebras
  • 3.2 The standard planar algebras of tangles and of chord diagrams and the no-go theorem
  • 3.3 Shielded chord diagrams and tangles
  • 3.4 Generators, relations and syzygies
• 4 Annular braids
• 5 Appendix: Symmetrization and Anonymization
  • 5.1 Linear substitution problems
  • 5.2 Symmetrization
  • 5.3 Anonymization
• 6 Appendix: Comments
  • 6.1 A comment on Figure 1
• Bibliography
• About this Paper...