AKT-09/HW3

DBN: Classes: 2009-10: AKT-09/Navigation # Week of... Videos, Notes, and Links 1 Sep 7 About This Class  090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.  090910-2: 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.  090917-1: The definition of finite type, weight systems, Jones is a finite type series.  090917-2: The skein relation for Jones; HOMFLY-PT 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.  090924-1: Some dimensions of ${\mathcal {A}}_{n}$, ${\mathcal {A}}$ is a commutative algebra, ${\mathcal {A}}(\bigcirc )\equiv {\mathcal {A}}(\uparrow )$. Class Photo  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  090929: The Milnor-Moore theorem, primitives, the map ${\mathcal {A}}^{r}\to {\mathcal {A}}$.  091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.  091001-2: The very basics on Lie algebras. 5 Oct 5  091006: Lie algebraic weight systems, $gl_{N}$.  091008-1: More on $gl_{N}$, Lie algebras and the four colour theorem.  091008-2: 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  091022-1: The Stonehenge Story to IHX and STU.  091022-2: The Stonhenge Story: anomalies, framings, relation with physics. 8 Oct 26  091027: Knotted trivalent graphs and their chord diagrams.  091029-1: Zsuzsi Dancso on the Kontsevich Integral (1).  091029-2: Zsuzsi Dancso on the Kontsevich Integral (2). 9 Nov 2  091103: The details of ${\mathcal {A}}^{TG}$.  091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.  091105-2: 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  091119-1: Local Khovanov homology, I.  091119-2: Local Khovanov homology, II. 12 Nov 23  091124: Emulation of one structure inside another, deriving the pentagon.  091126-1: Peter Lee on braided monoidal categories, I.  091126-2: Peter Lee on braided monoidal categories, II. 13 Nov 30  091201: The relations in KTG.  091203-1: The Existence of the Exponential Function.  091203-2: 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 Add your name / see who's in!

Solve the following problems and submit them in class by November 24, 2009:

Problem 1 With $T$, $B^{+}$ and $R$ as below, write $B^{+}$ as a composition of $T$ and two $R$'s, using he basic TG operations $d_{e}$, $u_{e}$, and $\#$.

Problem 2 Show that the "topological boundary" operator $\partial _{T}$ and the "crossing change" operator $x_{T}$ of the class of November 5 are compositions of the basic TG operations $d_{e}$, $u_{e}$, and $\#$ (you are also allowed to use "nullary" operations, otherwise known as "constants").

Problem 3 Write the third Reidemeister move R3 as a relation on $Z(B_{+})$.