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The definition of finite type, weight systems, Jones is a finite type series.
# | Week of... | Videos, Notes, and Links |
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1 | Sep 7 | About This Class 090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket. 090910-2: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial. Tricolourability |
2 | Sep 14 | 090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda. 090917-1: The definition of finite type, weight systems, Jones is a finite type series. 090917-2: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones. |
3 | Sep 21 | 090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots. 090924-1: Some dimensions of , is a commutative algebra, . Class Photo 090924-2: 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 090929: The Milnor-Moore theorem, primitives, the map . 091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T. 091001-2: The very basics on Lie algebras. |
5 | Oct 5 | 091006: Lie algebraic weight systems, . 091008-1: More on , Lie algebras and the four colour theorem. 091008-2: The "abstract tenssor" approach to weight systems, and PBW, the map . |
6 | Oct 12 | 091013: Algebraic properties of vs. algebraic properties of . Thursday's class canceled. |
7 | Oct 19 | 091020: Universal finite type invariants, filtered and graded spaces, expansions. Homework Assignment 2 The Stonehenge Story 091022-1: The Stonehenge Story to IHX and STU. 091022-2: The Stonhenge Story: anomalies, framings, relation with physics. |
8 | Oct 26 | 091027: Knotted trivalent graphs and their chord diagrams. 091029-1: Zsuzsi Dancso on the Kontsevich Integral (1). 091029-2: Zsuzsi Dancso on the Kontsevich Integral (2). |
9 | Nov 2 | 091103: The details of . 091105-1: Three basic problems: genus, unknotting numbers, ribbon knots. 091105-2: The three basic problems and algebraic knot theory. |
10 | Nov 9 | 091110: Tangles and planar algebras, shielding and the generators of KTG. Homework Assignment 3 No Thursday class. |
11 | Nov 16 | Local Khovanov Homology 091119-1: Local Khovanov homology, I. 091119-2: Local Khovanov homology, II. |
12 | Nov 23 | 091124: Emulation of one structure inside another, deriving the pentagon. 091126-1: Peter Lee on braided monoidal categories, I. 091126-2: Peter Lee on braided monoidal categories, II. |
13 | Nov 30 | 091201: The relations in KTG. 091203-1: The Existence of the Exponential Function. 091203-2: The Final Exam, Dror's failures. |
F | Dec 7 | The Final Exam on Thu Dec 10, 9-11, Bahen 6183. |
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(See also Vassiliev's Knot Invariants and Introduction to Vassiliev Knot Invariants.)
This is analogous to taking the first derivative.
where the sum is over the resolutions of and is the number of under resolutions made in obtaining .
Parallel to the result in calculus that partial derivatives commute, we have that is independent of the order in which the double points are resolved.
Notation: We drop the superscript in since for each , is only defined for -singular knots.
We can also express the 'type ' condition as:
whenever we have more than double points.
Given of type , We have .
Since , does not distinguish over crossing and under crossings in .
Let Failed to parse (unknown function\overcrossing): \mathcal{D}_m = \mathcal{K}_m / (\overcrossing=\undercrossing) .
Hence, the weight system given by is well-defined.