07-1352/Class Notes for January 30

07-1352/Navigation Panel   Back to 06-1350 # Week of... Links 1 Jan 9 About, Notes 2 Jan 16 Notes 3 Jan 23 Notes 4 Jan 30 Notes and Plans 5 Feb 6 Notes 6 Feb 13 Notes R Feb 20 Reading week 7 Feb 27 Notes 8 Mar 6 Notes 9 Mar 13 Notes 10 Mar 20 Notes 11 Mar 27 Notes 12 Apr 3 Map 13 Apr 10 Dancso SP Apr 17 Chu, Sankaran, Amir-Khosravi Suggested Student Topics

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 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

A construction of $Z$ following Kontsevich's Knizhnik-Zamolodchikov 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 ${\mathcal {B}}$ 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.