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

# 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
: 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.
Register of Good Deeds / To Do List

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0:00:33  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  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  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  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  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  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  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  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  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  Examples of the mapping $j_n$ on chord diagrams corresponding to $n$-singular knots (i.e. $W_{j_n}$ on $\mathcal{D}_m$)