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

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==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 in the subject. There are some gaps in my own knowledge of the subject; if you don't read on your own, at the end you'll have the same ones too.


===Today===
===Today===
Line 9: Line 11:
* 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}}.
The 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.
* The 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==

Revision as of 10:24, 30 January 2007

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In Preparation

The information below is preliminary and cannot be trusted! (v)

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 in the subject. There are some gaps in my own knowledge of the subject; if you don't read on your own, at the end you'll have the same ones too.

Today

  • A discussion of grading policy.
  • A bit more about quotients, and especially, some 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

The 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.

  • The 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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