AKT-09/HW2

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 October 29, 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 co-product 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.

1. Find a concise algorithm to compute the weight system $W_{so}$ associated with the Lie algebra $so(N)$ in its defining representation.
2. 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) 417-471.