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Video AKT-090917-2

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The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.

# Week of... Videos, Notes, and Links
1 Sep 7 About This Class
dbnvp 090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
dbnvp 090910-2: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
Tricolourability
2 Sep 14 dbnvp 090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda.
dbnvp 090917-1: The definition of finite type, weight systems, Jones is a finite type series.
dbnvp 090917-2: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
3 Sep 21 dbnvp 090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
dbnvp 090924-1: Some dimensions of {\mathcal A}_n, {\mathcal A} is a commutative algebra, {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow).
Class Photo
dbnvp 090924-2: {\mathcal A} is a co-commutative algebra, the relation with products of invariants, {\mathcal A} is a bi-algebra.
4 Sep 28 Homework Assignment 1
Homework Assignment 1 Solutions
dbnvp 090929: The Milnor-Moore theorem, primitives, the map {\mathcal A}^r\to{\mathcal A}.
dbnvp 091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
dbnvp 091001-2: The very basics on Lie algebras.
5 Oct 5 dbnvp 091006: Lie algebraic weight systems, gl_N.
dbnvp 091008-1: More on gl_N, Lie algebras and the four colour theorem.
dbnvp 091008-2: The "abstract tenssor" approach to weight systems, {\mathcal U}({\mathfrak g}) and PBW, the map {\mathcal T}_{\mathfrak g}.
6 Oct 12 dbnvp 091013: Algebraic properties of {\mathcal U}({\mathfrak g}) vs. algebraic properties of {\mathcal A}.
Thursday's class canceled.
7 Oct 19 dbnvp 091020: Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
dbnvp 091022-1: The Stonehenge Story to IHX and STU.
dbnvp 091022-2: The Stonhenge Story: anomalies, framings, relation with physics.
8 Oct 26 dbnvp 091027: Knotted trivalent graphs and their chord diagrams.
dbnvp 091029-1: Zsuzsi Dancso on the Kontsevich Integral (1).
dbnvp 091029-2: Zsuzsi Dancso on the Kontsevich Integral (2).
9 Nov 2 dbnvp 091103: The details of {\mathcal A}^{TG}.
dbnvp 091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.
dbnvp 091105-2: The three basic problems and algebraic knot theory.
10 Nov 9 dbnvp 091110: Tangles and planar algebras, shielding and the generators of KTG.
Homework Assignment 3
No Thursday class.
11 Nov 16 Local Khovanov Homology
dbnvp 091119-1: Local Khovanov homology, I.
dbnvp 091119-2: Local Khovanov homology, II.
12 Nov 23 dbnvp 091124: Emulation of one structure inside another, deriving the pentagon.
dbnvp 091126-1: Peter Lee on braided monoidal categories, I.
dbnvp 091126-2: Peter Lee on braided monoidal categories, II.
13 Nov 30 dbnvp 091201: The relations in KTG.
dbnvp 091203-1: The Existence of the Exponential Function.
dbnvp 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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0:00:33 [edit] We study the Jones polynomial for a double point and as a result get the Jones skein relation:

Failed to parse (unknown function\overcrossing): qJ(\overcrossing)-q^{-1}J(\undercrossing) = (q^{1/2}-q^{-1/2})J(\smoothing)
0:07:22 [edit] The Jones skein relation (discovered before the Kauffman bracket), together with the relation below, allows us to compute the Jones polynomial of any knot.

J(\bigcirc^k)=(q^{1/2} + q^{-1/2})^{k-1}
0:09:48 [edit] Definition: A knot diagram is descending if there exists a point on the diagram such that, starting from that point and going along the knot, each time we reach any crossing for the first time we do so along the upper strand of the crossing.

By flipping crossings one can make any knot descending and any descending knot is the unknot. From this we can deduce that the Jones skein relation and the value of J for the union of any number unknots give J explicitly for any knot.

0:11:52 [edit] Cousins of Jones skein relation: Conway-Alexander polynomial and HOMFLY-PT polynomial and basically any such relation (of this sort) you can write down defines a knot invariant.
0:18:40 [edit] Remark: The HOMFLY-PT polynomial contains both the Jones (set N=2) and the Conway-Alexander polynomial (take \partial/\partial N|_{N=0}).
0:20:43 [edit] Proof of the theorem from last hour: j_{(n, \cdot)} is a type n invariant. (in fact we may prove the same for both the Conway and HOMFLY-PT polynomial) Substitute q=e^x into the Jones skein relation, simplify J( double point ) and deduce the value of J on a n+1 singular knot \Rightarrow j_n vanishes on \mathcal{K}_{n+1}.
0:27:14 [edit] Rearranging the skein relation, we see that:

Failed to parse (unknown function\doublepoint): J(\doublepoint)=x(...)

Analogously,

Failed to parse (unknown function\doublepoint): J(\doublepoint ... \doublepoint)=x^{n+1}(...)

for n+1 double points. In particular, the coefficient of x^n, J_n, will be zero when evaluated on a knot with n+1 double points.

0:31:04 [edit] Computing W_{j_n} ( the weight system of type-n invariant j_n ) i.e. we need to figure out the constant term of the missing factor in J of n-singular knots.
0:39:12 [edit] Lemma: For any link L_k with k components, we have:

J_0(L_k)=J_0(\bigcirc^k)=2^{k-1}.
0:45:49 [edit] Examples of the mapping j_n on chord diagrams corresponding to n-singular knots (i.e. W_{j_n} on \mathcal{D}_m)