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

# Video AKT-090922

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Videography by Karene Chu

Notes on AKT-090922:    [edit, refresh]

FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.

# 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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0:00:02  Remark: Fixing the value of Jones polynomial on the unknot (instead of $k$ disjoint unknots, as given in last class) is sufficient. i.e. $J(O^k)=(q-q^{-1})J(O^{k-1})/(q^{1/2}-q^{-1/2})$ + induction.
0:02:41  Convention: $J(\bigcirc)=1$.
0:03:37  Notation: 'overcrossing' = $L_+$, 'undercrossing' = $L_-$ and 'smoothing' = $L_0$

(This is what I'm going to use from this point on, makes my job a lot easier! :-P )

0:03:53  Introducing the Alexander polynomial, (one of the many) definition:

1. $A(O^k) = \delta_{k,1}$
2. $A(L_+)-A(L_-)=z A(L_0)$

i.e. it's a special case of the Conway polynomial satisfying a special parametrization.

0:08:20  Theorem (from last time) If we expand $J(K)(e^x)=\sum{J_n(K)x^n}$ then $J_n$ is an invariant of type $n$.

In fact, this holds if we substitute $e^x$ by any power series that starts with $1+x$ (e.g. $1+x$, $1+sinx$).

0:09:27 
• The original statement Dror made here was '$W_{j_n}$ of any chord diagram is, you EITHER replace a chord with a bridge OR you replace it with $-2$ times a nothing'

I think a more accurate wording would be '$W_{j_n}$ of any chord diagram $= W_{j_n}$ of the diagram with a chord replaced with a bridge $- 2 \times W_{j_n}$ of the diagram with the same chord removed'.

0:10:13  Driving the weight system for the power series coefficients $h_n$ of the HOMFLY polynomial. i.e. $W_{h_n}$ of any chord diagram is $W_{h_n}$ of the diagram with a chord replaced by a bridge $- N \times W_{h_n}$ with the same chord deleted; $W_{h_n}(O^k) = N^{k-1}$.
0:13:31  Recall a consequence of having finitely many weight systems for type $n$ invariants is that we have only finitely many type $n$ invariants. But how many? Is it true that every map $\mathcal{D}_n \rightarrow A$ can be achieved as a weight system? (No)
0:14:59  Claim:

1. Every weight system $W_V$ sends any chord diagram with an isolated chord to $0$.
2. 4T relation
0:20:42  Proof of the claim -- see pictures. (I found it easier to work backwards from the expression with two double points so that the terms cancel out nicely)
0:30:13  Theorem: Any $W$ satisfying the FI and 4T relations is achieved as a weight system.
0:32:32  Check that $W_{h_n}$ satisfies the FI and 4T relations.
0:41:41  Definition: A framed knot is a smooth knot with a choice of a non-zero section of the normal bundle (mod homotopy).

• Conan's remark: for the ribbon definition, it is important that the ribbon has to be an orientable surface. i.e. a Mobius loop cannot the ribbon associated to a framed knot.
0:45:17  We have the short exact sequence:

$0 \rightarrow \mathbb{Z} \rightarrow \{$framed knots$\} \rightarrow \{$knots$\} \rightarrow 0$

with a splitting on the left given by the writhe.

0:49:28  The canonical $0$-framing of knots in $\mathbb{R}^3$ (Note that such canonical identification only exist in $\mathbb{R}^3$.
0:49:35  Remark: There is a Reidemeister theory for framed knots as well, where the R2 and R3 moves are allowed but not the R1 move. The writhe is an invariant of such knots.
0:51:57  Analogously, there are finite type invariants for framed knots and we have:

Theorem: Any $W$ satisfying the 4T relation comes from an invariant $V$ of framed knots.