**
Dror Bar-Natan
and
Dylan P. Thurston
**

*Date:*
This Edition: Jul. 23, 2001;
Revision History: Section 6.1.

We discuss a number of topics related to algebraic
constructions of universal finite type invariants. The idea is to find
presentations of knot theory, or of some mild generalizations of knot
theory, in terms of finitely many generators and relations, and then to
construct a universal finite type invariant by setting its values on the
generators so as the relations are satisfied. One such presentation
involves knotted trivalent graphs, and is genuinely 3-dimensional. In
this presentation the generators turns out to be the standardly
embedded tetrahedron
and the relations are on one hand
equivalent to the pentagon and hexagon relations of Drinfel'd's theory
of associators and on the other hand they are closely related to the
Biedenharn-Elliot identities of -symbols and to the Pachner moves of
the theory of triangulations. Another such presentation involves Jones'
notion of a planar algebra [J] and leads to a
crossing-centric constructions of a universal finite type invariant (as
opposed to the now-standard associativity-centric construction). Much
of what we discuss is work in progress, and this article contains many
``live ends'', unfinished problems that don't seem to be dead ends.

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- Contents
- 1 Introduction
- 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 -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
- 4 Annular braids
- 5 Appendix: Symmetrization and Anonymization
- 6 Appendix: Comments
- Bibliography
- About this Paper...