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Video AKT-090917-1

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The definition of finite type, weight systems, Jones is a finite type series.

# 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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0:00:00 [edit] Today and for a little further we will be following my 1992 paper, On the Vassiliev Knot Invariants.

(See also Vassiliev's Knot Invariants and Introduction to Vassiliev Knot Invariants.)

0:00:01 [edit] Comments about the website (Correction: Conan is in northWESTERN!)
0:01:03 [edit] Computing the tricolouring invariant efficiently: A priori, it is an exponential time problem since for a knot diagram with n arcs we need to consider 3^n possible colourings. However, the problem can be reduced to Gaussian elimination over Z/3Z and quadratic time. For more details see AKT-09/Tricolourability.
0:05:25 [edit] Correction on last time's correction on Jones polynomial: Instead of substituting A=q^{1/4}, we should substitute A= i q^{-1/4} (For knots, this is same as setting q = A^{-4} since A only appear in terms of A^4)
0:08:22 [edit] Universal polynomial invariant: Looking at polynomials defined on the space of knots
0:10:01 [edit] What is a polynomial: Definition for both continuous and discrete spaces (higher derivatives or differences vanish)
0:11:09 [edit] Defining 'derivative' for maps from the space of oriented n-singular knots embedded in an oriented copy of \mathbb{R}^3 to Abelian group A. (Given knot invariant V, V' maps n-singular knots to sums of n-1-singular knots and V^{(m)} maps m-singular knots to A )
0:11:19 [edit] Let \mathcal{K} denote the space of oriented knots in an oriented \mathbb{R}^3 and A be any abelian group. Then, given any invariant V: \mathcal{K} \rightarrow A, we can extend V to 1-singular knots (i.e. knots with one double point) by setting:

Failed to parse (unknown function\doublepoint): V^{(1)}(\doublepoint)=V(\overcrossing) - V(\undercrossing)


This is analogous to taking the first derivative.

0:15:48 [edit] Likewise, for an m-singular knot K, we can define:

V^{(m)}(K)=\sum_{K'}{(-1)^{u(K')}V(K')}

where the sum is over the 2^m resolutions of K and u(K') is the number of under resolutions made in obtaining K'.

Parallel to the result in calculus that partial derivatives commute, we have that V^{(m)}(K) is independent of the order in which the double points are resolved.

0:20:08 [edit] Definition: A knot invariant V is of Vassiliev type m if V^{(m+1)} = 0 (on the whole space of (m+1)-singular knots).

Notation: We drop the superscript in V^{(m)} since for each m, V^{(m)} is only defined for m-singular knots.

We can also express the 'type m' condition as:

Failed to parse (unknown function\doublepoint): V(\doublepoint ... \doublepoint)=0


whenever we have more than m double points.

0:22:10 [edit] Motivation: instead of studying the knot invariant itself, we may study the derivatives of it.
0:23:37 [edit] Let \mathcal{K}_m = \{ m-singular knots \}

Given V of type m, We have V^{(m)}: \mathcal{K}_m \rightarrow A.

Since V^{(m+1)}=0, V^{(m)} does not distinguish over crossing and under crossings in \mathcal{K}_m.

Let Failed to parse (unknown function\overcrossing): \mathcal{D}_m = \mathcal{K}_m / (\overcrossing=\undercrossing) .

Hence, the weight system  \mathcal{D}_m \rightarrow A given by W_V = V^{(m)} is well-defined.

0:28:15 [edit] Claim: \mathcal{D}_m corresponds to the set of chord diagrams. (Deduce that there are at most finitely many linearly independent invariants of type m.)
0:31:51 [edit] Proof of the claim: Walking along the knot and mark the singular points.
0:46:11 [edit] Let K be a knot, J_K(q) be its Jones polynomial. Substitute q = e^x and expand J_k(e^x) into power series. We have J_K(e^x) = \sum_n j_{(n,K)} \ x^n where the coefficients j_{(n,\cdot)}: \{knots \} \rightarrow \mathbb{Z} are knot invariants.

Thm: j_{(n, \cdot)} is of type n.