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3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
|#||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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The arcs of can be coloured in 3 colours s.t.
Terminology (for the purpose of this annotation)
Local rule: at each crossing, of the 3 arcs involved, either 1 or all 3 colours appear.
Global rule: all 3 colours must appear.
(Together with the provision of 3 colours, knots which can be coloured obeying these rules, are called tricolourable as defined at 0:10:26 of this hour.)
Problem/Concern: Sometimes, in a local picture, only 2 colours appear on one side of an isotopy move (e.g. R2) whereas all 3 colours appear on the other. One might worry that this could lead to the violation of the global rule.
Solution: One can prove that a knot (consisting of only 1 connected piece of material in ), which is coloured obeying the local rule of tricolourability, has at least 2 colours it has all 3 colours. The proof relies on the fact that the same piece of material can change colour (from one colour to a 2nd colour) only by going 'under' a crossing, and whenever a crossing involves 2 colours it must involve a 3rd. (This argument fails, however, for links. Just consider two knots, one red and one blue say, placed side by side.)
These suffice because for any knot or link diagram we can apply the first relation recursively until we eliminate all the crossings and end up with a disjoint union of unknots. For instance, we get:
Failed to parse (unknown function\HopfLink): \left\langle \HopfLink \right\rangle = A^2d^2 + 2ABd + B^2d^2