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Week of...

Videos, Notes, and Links

1

Sep 7

About This Class 0909101: 3colourings, Reidemeister's theorem, invariance, the Kauffman bracket. 0909102: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial. Tricolourability

2

Sep 14

090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda. 0909171: The definition of finite type, weight systems, Jones is a finite type series. 0909172: The skein relation for Jones; HOMFLYPT and Conway; the weight system of Jones.

3

Sep 21

090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots. 0909241: Some dimensions of ${\mathcal {A}}_{n}$, ${\mathcal {A}}$ is a commutative algebra, ${\mathcal {A}}(\bigcirc )\equiv {\mathcal {A}}(\uparrow )$. Class Photo 0909242: ${\mathcal {A}}$ is a cocommutative algebra, the relation with products of invariants, ${\mathcal {A}}$ is a bialgebra.

4

Sep 28

Homework Assignment 1 Homework Assignment 1 Solutions 090929: The MilnorMoore theorem, primitives, the map ${\mathcal {A}}^{r}\to {\mathcal {A}}$. 0910011: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T. 0910012: The very basics on Lie algebras.

5

Oct 5

091006: Lie algebraic weight systems, $gl_{N}$. 0910081: More on $gl_{N}$, Lie algebras and the four colour theorem. 0910082: The "abstract tenssor" approach to weight systems, ${\mathcal {U}}({\mathfrak {g}})$ and PBW, the map ${\mathcal {T}}_{\mathfrak {g}}$.

6

Oct 12

091013: Algebraic properties of ${\mathcal {U}}({\mathfrak {g}})$ vs. algebraic properties of ${\mathcal {A}}$. Thursday's class canceled.

7

Oct 19

091020: Universal finite type invariants, filtered and graded spaces, expansions. Homework Assignment 2 The Stonehenge Story 0910221: The Stonehenge Story to IHX and STU. 0910222: The Stonhenge Story: anomalies, framings, relation with physics.

8

Oct 26

091027: Knotted trivalent graphs and their chord diagrams. 0910291: Zsuzsi Dancso on the Kontsevich Integral (1). 0910292: Zsuzsi Dancso on the Kontsevich Integral (2).

9

Nov 2

091103: The details of ${\mathcal {A}}^{TG}$. 0911051: Three basic problems: genus, unknotting numbers, ribbon knots. 0911052: The three basic problems and algebraic knot theory.

10

Nov 9

091110: Tangles and planar algebras, shielding and the generators of KTG. Homework Assignment 3 No Thursday class.

11

Nov 16

Local Khovanov Homology 0911191: Local Khovanov homology, I. 0911192: Local Khovanov homology, II.

12

Nov 23

091124: Emulation of one structure inside another, deriving the pentagon. 0911261: Peter Lee on braided monoidal categories, I. 0911262: Peter Lee on braided monoidal categories, II.

13

Nov 30

091201: The relations in KTG. 0912031: The Existence of the Exponential Function. 0912032: The Final Exam, Dror's failures.

F

Dec 7

The Final Exam on Thu Dec 10, 911, Bahen 6183.

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Solve the following problems and submit them in class by November 3, 2009:
Problem 1. Let ${\mathfrak {g}}_{1}$ and ${\mathfrak {g}}_{2}$ be finite dimensional metrized Lie algebras, let ${\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2}$ denote their direct sum with the obvious "orthogonal" bracket and metric, and let $m$ be the canonical isomorphism $m:{\mathcal {U}}({\mathfrak {g}}_{1})\otimes {\mathcal {U}}({\mathfrak {g}}_{2})\to {\mathcal {U}}({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})$. Prove that
${\mathcal {T}}_{{\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2}}=m\circ ({\mathcal {T}}_{{\mathfrak {g}}_{1}}\otimes {\mathcal {T}}_{{\mathfrak {g}}_{2}})\circ \Box$,
where $\Box :{\mathcal {A}}(\uparrow )\to {\mathcal {A}}(\uparrow )\otimes {\mathcal {A}}(\uparrow )$ is the coproduct and ${\mathcal {T}}_{\mathfrak {g}}$ denotes the ${\mathcal {U}}({\mathfrak {g}})$valued "tensor map" on ${\mathcal {A}}$. Can you relate this with the first problem of HW1?
Problem 2.
 Find a concise algorithm to compute the weight system $W_{so}$ associated with the Lie algebra $so(N)$ in its defining representation.
 Verify that your algorithm indeed satisfies the $4T$ relation.
Problem 3. The Kauffman polynomial $F(K)(a,z)$ (see [Kauffman]) of a knot or link $K$ is $a^{w(K)}L(K)$ where $w(L)$ is the writhe of $K$ and where $L(K)$ is the regular isotopy invariant defined by the skein relations
$L(s_{\pm })=a^{\pm 1}L(s))$
(here $s$ is a strand and $s_{\pm }$ is the same strand with a $\pm$ kink added) and
Failed to parse (unknown function "\backoverslash"): {\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)}
and by the initial condition $L(\bigcirc )=1$. State and prove the relationship between $F$ and $W_{so}$.
Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.
[Kauffman] ^ L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417471.