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 [math]\displaystyle{ {\mathcal A}_n }[/math], [math]\displaystyle{ {\mathcal A} }[/math] is a commutative algebra, [math]\displaystyle{ {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow) }[/math]. Class Photo  090924-2: [math]\displaystyle{ {\mathcal A} }[/math] is a co-commutative algebra, the relation with products of invariants, [math]\displaystyle{ {\mathcal A} }[/math] is a bi-algebra. 4 Sep 28 Homework Assignment 1 Homework Assignment 1 Solutions  090929: The Milnor-Moore theorem, primitives, the map [math]\displaystyle{ {\mathcal A}^r\to{\mathcal A} }[/math].  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, [math]\displaystyle{ gl_N }[/math].  091008-1: More on [math]\displaystyle{ gl_N }[/math], Lie algebras and the four colour theorem.  091008-2: The "abstract tenssor" approach to weight systems, [math]\displaystyle{ {\mathcal U}({\mathfrak g}) }[/math] and PBW, the map [math]\displaystyle{ {\mathcal T}_{\mathfrak g} }[/math]. 6 Oct 12  091013: Algebraic properties of [math]\displaystyle{ {\mathcal U}({\mathfrak g}) }[/math] vs. algebraic properties of [math]\displaystyle{ {\mathcal A} }[/math]. 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 [math]\displaystyle{ {\mathcal A}^{TG} }[/math].  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 3, 2009:

Problem 1. Let [math]\displaystyle{ {\mathfrak g}_1 }[/math] and [math]\displaystyle{ {\mathfrak g}_2 }[/math] be finite dimensional metrized Lie algebras, let [math]\displaystyle{ {\mathfrak g}_1\oplus{\mathfrak g}_2 }[/math] denote their direct sum with the obvious "orthogonal" bracket and metric, and let [math]\displaystyle{ m }[/math] be the canonical isomorphism [math]\displaystyle{ m:{\mathcal U}({\mathfrak g}_1)\otimes{\mathcal U}({\mathfrak g}_2)\to{\mathcal U}({\mathfrak g}_1\oplus{\mathfrak g}_2) }[/math]. Prove that

[math]\displaystyle{ {\mathcal T}_{{\mathfrak g}_1\oplus{\mathfrak g}_2} = m\circ({\mathcal T}_{{\mathfrak g}_1}\otimes{\mathcal T}_{{\mathfrak g}_2})\circ\Box }[/math],

where [math]\displaystyle{ \Box:{\mathcal A}(\uparrow)\to{\mathcal A}(\uparrow)\otimes{\mathcal A}(\uparrow) }[/math] is the co-product and [math]\displaystyle{ {\mathcal T}_{{\mathfrak g}} }[/math] denotes the [math]\displaystyle{ {\mathcal U}({\mathfrak g}) }[/math]-valued "tensor map" on [math]\displaystyle{ {\mathcal A} }[/math]. Can you relate this with the first problem of HW1?

Problem 2.

1. Find a concise algorithm to compute the weight system [math]\displaystyle{ W_{so} }[/math] associated with the Lie algebra [math]\displaystyle{ so(N) }[/math] in its defining representation.
2. Verify that your algorithm indeed satisfies the [math]\displaystyle{ 4T }[/math] relation.

Problem 3. The Kauffman polynomial [math]\displaystyle{ F(K)(a,z) }[/math] (see [Kauffman]) of a knot or link [math]\displaystyle{ K }[/math] is [math]\displaystyle{ a^{-w(K)}L(K) }[/math] where [math]\displaystyle{ w(L) }[/math] is the writhe of [math]\displaystyle{ K }[/math] and where [math]\displaystyle{ L(K) }[/math] is the regular isotopy invariant defined by the skein relations

[math]\displaystyle{ L(s_\pm)=a^{\pm 1}L(s)) }[/math]

(here [math]\displaystyle{ s }[/math] is a strand and [math]\displaystyle{ s_\pm }[/math] is the same strand with a [math]\displaystyle{ \pm }[/math] kink added) and

[math]\displaystyle{ L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right) }[/math]

and by the initial condition [math]\displaystyle{ L(\bigcirc)=1 }[/math]. State and prove the relationship between [math]\displaystyle{ F }[/math] and [math]\displaystyle{ W_{so} }[/math].

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.