071352/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:
 More about quotients, and especially, a discussion of The HOMFLY Braidor Algebra.
 Ng's nogo theorem for ribbon knots (arXiv:qalg/9502017 and arXiv:math.GT/0310074) and the nogo theorems we still don't have (and with a lot of luck, we'll never have) about the fusion number, the unknotting number and the genus.
The Following Two Weeks  Weeks 5 and 6
A construction of following Kontsevich's KnizhnikZamolodchikov approach with the added wisdom of [Le_Murakami_97] and [Murakami_Ohtsuki_97].
Week 7 and 8
Back to the PBW theorem.
 Proof and some variants.
 Wheeling and the algebra morphism from to .
Weeks 9 and 10
Back to the degreebydegree approach.
 A construction of using parenthesized tangles.
 "Gauge equivalence" and the near uniqueness of .
 What this all means in the language of knotted trivalent graphs.
Weeks 11 through 13
Reserved.
 Students lectures?
 More on the relationship with ChernSimons theory?
 LMO, Århus and invariants of 3manifolds?
 Somethings else?
References
[Le_Murakami_97] ^ T. Q. T. Le and J. Murakami, Parallel Version of the Universal VassilievKontsevich Invariant, Journal of Pure and Applied Algebra 121 (1997) 271291.
[Murakami_Ohtsuki_97] ^ J. Murakami and T. Ohtsuki, Topological Quantum Field Theory for the Universal Quantum Invariant, Communications in Mathematical Physics 1883 (1997) 501520.