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Video AKT-091008-1

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

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

# 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.
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0:00:00 [edit] In part of today's class we will be sketching Dror's paper Lie Algebras and the Four Color Theorem.
0:00:12 [edit] 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 [edit] Completing the computation on gl_n from last time: Graphical interpretation of factors associated to each diagram part.
0:07:24 [edit] Completing the computation for intersection with a chord from the circle.
0:08:44 [edit] Evaluate the functional on the diagram with a single chord
0:12:56 [edit] Sample diagram 2: circle with two crossing chords

Sample diagram 3: circle with two non-crossing chords

0:14:19 [edit] 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 [edit] Verify the 4T relation
0:19:04 [edit] 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 [edit] 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 [edit] 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)).

0:31:58 [edit] Comments:

  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 [edit] Translate of the statement:

Every planar diagram has a 4-coloring.

0:38:26 [edit] 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 [edit] Establish relation between highest power of N in W_{gl_N}(D) and the genus of the surface obtained.
0:46:33 [edit] Conclusion: highest power is \deg(D)+2 iff genus =0 i.e. the graph is planner
0:46:51 [edit] Sketch for claim 2: (via tensors) obtain W_{sl_2}(D) being the number of 3 colorings of the edges.
0:51:07 [edit] 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.