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Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
|#||Week of...||Videos, Notes, and Links|
|1||Sep 7||About This Class|
: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
|2||Sep 14||: More on Jones, some pathologies and more on Reidemeister, our overall agenda.|
: The definition of finite type, weight systems, Jones is a finite type series.
: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
|3||Sep 21||: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.|
: Some dimensions of , is a commutative algebra, .
: is a co-commutative algebra, the relation with products of invariants, is a bi-algebra.
|4||Sep 28||Homework Assignment 1|
Homework Assignment 1 Solutions
: The Milnor-Moore theorem, primitives, the map .
: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
: The very basics on Lie algebras.
|5||Oct 5||: Lie algebraic weight systems, .|
: More on , Lie algebras and the four colour theorem.
: The "abstract tenssor" approach to weight systems, and PBW, the map .
|6||Oct 12||: Algebraic properties of vs. algebraic properties of .|
Thursday's class canceled.
|7||Oct 19||: Universal finite type invariants, filtered and graded spaces, expansions.|
Homework Assignment 2
: The Stonehenge Story to IHX and STU.
: The Stonhenge Story: anomalies, framings, relation with physics.
|8||Oct 26||: Knotted trivalent graphs and their chord diagrams.|
: Zsuzsi Dancso on the Kontsevich Integral (1).
: Zsuzsi Dancso on the Kontsevich Integral (2).
|9||Nov 2||: The details of .|
: Three basic problems: genus, unknotting numbers, ribbon knots.
: The three basic problems and algebraic knot theory.
|10||Nov 9||: Tangles and planar algebras, shielding and the generators of KTG.|
Homework Assignment 3
No Thursday class.
: Local Khovanov homology, I.
: Local Khovanov homology, II.
|12||Nov 23||: Emulation of one structure inside another, deriving the pentagon.|
: Peter Lee on braided monoidal categories, I.
: Peter Lee on braided monoidal categories, II.
|13||Nov 30||: The relations in KTG.|
: The Existence of the Exponential Function.
: The Final Exam, Dror's failures.
|F||Dec 7||on Thu Dec 10, 9-11, Bahen 6183.|
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Space of knots ; space of finite type knot invariants ; weight systems of finite type invariants where
We proved that is a graded commutative co-commutative bi-algebra.
set generated by trivalent diagrams (AS, STU)
definition of IHX relation
Pf: essentially same as (close/open the loop, apply the relations to prove 'open' is well defined)
(By induction on the number of internal vertices)