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

# Video AKT-091008-1

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Videography by Karene Chu

Notes on AKT-091008-1:    [edit, refresh]

More on $gl_N$, Lie algebras and the four colour theorem.

# 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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0:00:00  In part of today's class we will be sketching Dror's paper Lie Algebras and the Four Color Theorem.
0:00:12  Reminder from last time: Given a Lie-algebra(metrized etc.) and a finite dimensional representation, we construct functional from $\mathcal{A}$ to $\mathbb{Q}$. We computed the map explicitly for the example $gl_n$.
0:03:55  Completing the computation on $gl_n$ from last time: Graphical interpretation of factors associated to each diagram part.
0:07:24  Completing the computation for intersection with a chord from the circle.
0:08:44  Evaluate the functional on the diagram with a single chord
0:12:56  Sample diagram 2: circle with two crossing chords

Sample diagram 3: circle with two non-crossing chords

0:14:19  Sample diagram 4: the $D_{ai}$ diagram

(The value being $n$ raised to the power equal to the number of connected components after we replace all intersection points according to the recipe.)

0:17:57  Verify the 4T relation
0:19:04  Comment: $W_{gl_N, R^N}$ is the HOMFLY weight system up to the framing independence relation. i.e.

$W_{gl_N, R^N} \circ p (D) = W_H(D)$
0:21:48  Aside: $W_{\mathcal{G}}$ (without the representation) makes sense on $\mathcal{A}(\phi)$ (trivalent diagrams without skeleton) with the option to mod out by the AS and IHX relations.
0:23:59  Statement:

" $W_{sl_2}(D)=0 \ \Rightarrow \ W_{sl_N}^{top}(D)=0$ "

where $W_{sl_N}^{top}(D) = \mbox{Coeff}_{N^{\deg{D}+2}}(W_{sl_N}(D))$.

1. The statement is reasonable and doesn't look hard.
2. $|W_{sl_N}^{top}(D)|$ is proportional to the number of planar embeddings of $D$ with minor conditions.
3. If $D$ is planar, $|W_{sl_2}(D)|$ is proportional to the number of 4-colorings of the plane divided by $D$.
4. The statement above is equivalent to the four colour theorem.
0:35:51  Translate of the statement:

Every planar diagram has a 4-coloring.

0:38:26  Sketch of the proof (Claim 1) Thicken the diagram $\rightarrow$ surface with boundary (with a copy of $D$ embedded) $\rightarrow$ glue discs along boundary components $\rightarrow$ closed surface with a copy of $D$
0:45:23  Establish relation between highest power of $N$ in $W_{gl_N}(D)$ and the genus of the surface obtained.
0:46:33  Conclusion: highest power is $\deg(D)+2$ iff genus $=0$ i.e. the graph is planner
0:46:51  Sketch for claim 2: (via tensors) obtain $W_{sl_2}(D)$ being the number of 3 colorings of the edges.
0:51:07  Given a planar diagram, number of 3-colorings of the edges = number of 4-colorings of the regions in the plane divided by the diagram.

Bijection using the Klein 4 group as colors and add when crossing edges.