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.
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 no-go theorem for ribbon knots (arXiv:q-alg/9502017 and arXiv:math.GT/0310074) and the no-go 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
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 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
- Students lectures?
- More on the relationship with Chern-Simons theory?
- LMO, Århus and invariants of 3-manifolds?
- Somethings else?
[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.