07-1352/Class Notes for January 30: Difference between revisions

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{{In Preparation}}


==Tentative Future Plans==
==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===
===Today===
* A discussion of grading policy.
A discussion of grading policy and of future plans, and then a long list of "I don't know"s regarding:
* A bit more about quotients, and especially, some discussion of [[The HOMFLY Braidor Algebra]].
* 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.
* 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===
===The Following Two Weeks - Weeks 5 and 6===
* The construction of Z following Kontsevich's Knizhnik-Zamolodchikov approach with the added wisdom of {{ref|Le_Murakami_97}} and {{ref|Murakami_Ohtsuki_97}}.
A construction of <math>Z</math> following Kontsevich's Knizhnik-Zamolodchikov approach with the added wisdom of {{ref|Le_Murakami_97}} and {{ref|Murakami_Ohtsuki_97}}.

===Week 7 and 8===
Back to the PBW theorem.
* Proof and some variants.
* Wheeling and the algebra morphism from <math>{\mathcal B}</math> to <math>{\mathcal A}</math>.

===Weeks 9 and 10===
Back to the degree-by-degree approach.
* A construction of <math>Z</math> using parenthesized tangles.
* "Gauge equivalence" and the near uniqueness of <math>Z</math>.
* 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==
==References==

Latest revision as of 15:57, 30 January 2007

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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 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 to .

Weeks 9 and 10

Back to the degree-by-degree 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 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.

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