07-1352/Class Notes for January 30

Tentative Future Plans

My overall plan is to dump on you unsuspecting victims all (or at least most) of what I know which is relevant to the construction of an Algebraic Knot Theory so that at the end of this class you will in principle be as ready as I am to carry out independent research on the subject. There are some gaps in my own knowledge; if you don't think on your own, at the end you'll have the same ones too.

Today

A discussion of grading policy and of future plans, and then a long list of "I don't know"s regarding:

The Following Two Weeks - Weeks 5 and 6

A construction of $Z$ following Kontsevich's Knizhnik-Zamolodchikov approach with the added wisdom of and .

Week 7 and 8

Back to the PBW theorem.

• Proof and some variants.
• Wheeling and the algebra morphism from ${\mathcal B}$ to ${\mathcal A}$.

Weeks 9 and 10

Back to the degree-by-degree approach.

• A construction of $Z$ using parenthesized tangles.
• "Gauge equivalence" and the near uniqueness of $Z$.
• What this all means in the language of knotted trivalent graphs.

Weeks 11 through 13

Reserved.

• Students lectures?
• More on the relationship with Chern-Simons theory?
• LMO, Århus and invariants of 3-manifolds?
• Somethings else?

References

[Le_Murakami_97] ^  T. Q. T. Le and J. Murakami, Parallel Version of the Universal Vassiliev-Kontsevich Invariant, Journal of Pure and Applied Algebra 121 (1997) 271-291.

[Murakami_Ohtsuki_97] ^  J. Murakami and T. Ohtsuki, Topological Quantum Field Theory for the Universal Quantum Invariant, Communications in Mathematical Physics 188-3 (1997) 501-520.