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

Video 090915

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Videography by Karene Chu

Notes on AKT-090915:    [edit, refresh]

More on Jones, some pathologies and more on Reidemeister, our overall agenda.

# 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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0:00:52 [edit] correction from last time: regarding 0-smoothing vs. 1-smoothing

(A way to remember the smoothings: Call the bottom strand 'level 0' and the top strand 'level 1'. Then, for the k-th smoothing enter the 'interchange' at level k and turn right).

0:03:24 [edit] Review and additions to last class, corrections:

  1. The Jones polynomial is usually normalized by diving by \left\langle \bigcirc \right\rangle, the bracket of the unknot (i.e. dividing by an additional factor of d).
  2. We can prove that for any knot K, J(K) is a polynomial of A^4. Hence, we can substitute A=q^{1/4} to get a Laurent polynomial in q.
0:08:07 [edit] How effective is the Jones polynomial in distinguishing knots with 10 or less crossings (250 knots with 243 different Jones polynomials)
0:09:00 [edit] This Mathematica notebook is Classes/09-1350/nb/AKT-090915-ThePowerOfJones.pdf.
0:10:37 [edit] Continuity is insufficient for defining 'nice' knots in \mathbb{R}^3: pathological examples when defining knots by continuous maps modulo homotopy (see for instance Wild Knots).
0:12:22 [edit] The handout shown is classes/0102/AlgTop/Pathologies/print.html.
0:15:01 [edit] Correct definition: smooth injections from S^1 to \mathbb{R}^3 mod smooth deformations of such maps. Sketch of a proof of Reidemeister's theorem via tubular neighborhoods and polygonal knots.
0:20:12 [edit] Administrative comments (non-math from this point on)
0:33:12 [edit] Ideology of the class and philosophical comments:

* View knot theory as an algebraic structure. Why?
  1. Be able to answer more sophisticated questions.
  2. Knot theory is finitely generated (if we use the right algebraic structure):
{knot theory}=\left\langle g_1,...,g_n| \; r_1, ...,r_m\right\rangle \longrightarrow a good target space
* We will study polynomials on \mathcal{K} (the space of knots).
0:47:57 [edit] Find a 'good' target space by looking at polynomials on the space of knots
0:50:27 [edit] The Seifert Algorithm drawings by Emily Redelmeier are from classes/0304/KnotTheory/SeifertAlgorithm/index.html.
0:51:05 [edit] The "Trefoil Knot is Fibered" animation is due to Robert Barrington Leigh, and it is available at People/BarringtonLeigh/FiberedKnot.html. (See also [Fibered knot].)