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Video 090910-2

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R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.

# 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.
Register of Good Deeds / To Do List
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0:00:08 [edit] Well-defineness of the Kauffman bracket: The resulting polynomial (or the resulting zoo of trivial links) is independent of the order of smoothing.
0:01:47 [edit] Is the Kauffman bracket a knot invariant? Need to check invariance under the 3 Reidemeister moves (assuming invariance under planar isotopy). It turns out that R2 yields the most interesting constraints on the variables A, B and d. So, we will proceed in the order R2, R3 then R1.
0:02:16 [edit] Invariance under R2 forces us to set (after computation and comparison):

B=A^{-1}
d=-A^2-A^{-2}

Now, the Kauffman bracket becomes a Laurent polynomial in a single variable, A.

0:08:28 [edit] Invariance under R3: automatically satisfied, as long as the above relations between the 3 variables are satisfied (i.e. invariant under R2).
0:11:41 [edit] Invariance under R1: get -A^3=1.
0:15:44 [edit] Definitions of positive and negative crossings and the writhe of an oriented knot diagram.

Positive crossing: Failed to parse (unknown function\overcrossing): \,\overcrossing\,
Negative crossing: Failed to parse (unknown function\undercrossing): \,\undercrossing\,
The writhe of an oriented knot diagram is the sum of the signs of all the crossings in the diagram.
0:19:35 [edit] Remark: For a (single component!) knot, reversing the orientation of the knot doesn't change the signs of the crossings and hence preserves the writhe (the orientation of both strands in each crossing is reversed).
0:21:36 [edit] The writhe is invariant under R2 and R3 but not under R1: the writhe of a knot diagram with an extra kink is the writhe of the original diagram \pm 1 depending on the sign of the added crossing.
0:26:46 [edit] The Jones polynomial: a clever combination of the Kauffman bracket and the writhe gives a knot invariant.

J(D)=\left\langle D\right\rangle \cdot (-A^3)^{w(D)}
0:27:51 [edit] If D' is the knot diagram D with a positive kink added, then

\left\langle D' \right\rangle =(-A^{-3})\cdot \left\langle D \right\rangle
(-A^3)^{w(D')}=(-A^3) \cdot (-A^3)^{w(D)}

Therefore, J(D')=J(D) and similarly for a negative kink, i.e. the Jones polynomial is invariant under R1.

0:30:00 [edit] The Jones polynomial is readily computable and stronger than tricolourability: For instance, we can use it to show that Left-hand Trefoil \neq Right-hand Trefoil.
0:33:32 [edit] A presentation for knot diagrams, see also Planar Diagrams.
0:40:11 [edit] Find this Mathematica notebook at Pensieve/2009-09 under "AKT-090910-ComputingJones". A further discussion of this program is at the Knot Atlas, including a MUCH more efficient version.
0:49:06 [edit] The Jones polynomial of the mirror of a knot in the variable A is the same as the Jones polynomial of the original knot with A replaced by A^{-1}:

J(\overline{K})(A)=J(K)(A^{-1})
0:51:07 [edit] Dror suggests two problems to think about:

1. How hard is it to compute I, the colouring invariant. (See AKT-09/Tricolourability for a proposed simple procedure.)
2. How powerful is the Jones polynomial? (See the next class for the answer.)