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

# Video 090910-2

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Notes on AKT-090910-2:    [edit, refresh]

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

# 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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0:00:08  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  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  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  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  Invariance under R1: get $-A^3=1$.
0:15:44  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  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  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  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  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  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.
0:49:06  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 $A^{-1}$:
$J(\overline{K})(A)=J(K)(A^{-1})$