© | Dror Bar-Natan: Classes: 2009-10: AKT:

# 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
: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
Tricolourability
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 ${\mathcal A}_n$, ${\mathcal A}$ is a commutative algebra, ${\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow)$.
Class Photo
: ${\mathcal A}$ 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
: The Milnor-Moore theorem, primitives, the map ${\mathcal A}^r\to{\mathcal A}$.
: 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, $gl_N$.
: More on $gl_N$, Lie algebras and the four colour theorem.
: The "abstract tenssor" approach to weight systems, ${\mathcal U}({\mathfrak g})$ and PBW, the map ${\mathcal T}_{\mathfrak g}$.
6 Oct 12 : Algebraic properties of ${\mathcal U}({\mathfrak g})$ vs. algebraic properties of ${\mathcal A}$.
Thursday's class canceled.
7 Oct 19 : Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
: 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 ${\mathcal A}^{TG}$.
: 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.
11 Nov 16 Local Khovanov Homology
: 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 The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
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0:00:00  Today and for a little further we will be following my 1992 paper, On the Vassiliev Knot Invariants.

0:00:01  Comments about the website (Correction: Conan is in northWESTERN!)
0:01:03  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  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  Universal polynomial invariant: Looking at polynomials defined on the space of knots
0:10:01  What is a polynomial: Definition for both continuous and discrete spaces (higher derivatives or differences vanish)
0:11:09  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  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  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  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  Motivation: instead of studying the knot invariant itself, we may study the derivatives of it.
0:23:37  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  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  Proof of the claim: Walking along the knot and mark the singular points.
0:46:11  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 $n$.