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Video AKT-090922

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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
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.
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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0:00:02 [edit] 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 [edit] Convention: J(\bigcirc)=1.
0:03:37 [edit] 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 [edit] 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 [edit] 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 [edit]

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 [edit] 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 [edit] 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 [edit] Claim:

  1. Every weight system W_V sends any chord diagram with an isolated chord to 0.
  2. 4T relation
0:20:42 [edit] 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 [edit] Theorem: Any W satisfying the FI and 4T relations is achieved as a weight system.
0:32:32 [edit] Check that W_{h_n} satisfies the FI and 4T relations.
0:41:41 [edit] Definition: A framed knot is a smooth knot with a choice of a non-zero section of the normal bundle (mod homotopy).

0:45:17 [edit] 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 [edit] The canonical 0-framing of knots in \mathbb{R}^3 (Note that such canonical identification only exist in \mathbb{R}^3.
0:49:35 [edit] 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 [edit] 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.