AKT-09/HW2: Difference between revisions

Revision as of 12:14, 19 October 2009

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.
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In Preparation

The information below is preliminary and cannot be trusted! (v)

Solve the following problems and submit them in class by October 29, 2009:

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

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 {\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 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 \Box:{\mathcal A}(\uparrow)\to{\mathcal A}(\uparrow)\otimes{\mathcal A}(\uparrow)} is the co-product 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 {\mathcal T}_{{\mathfrak g}}} denotes the 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 {\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 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 W_{so}} associated with the Lie algebra 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 so(N)} in its defining representation.
Verify that your algorithm indeed satisfies the 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 4T} relation.

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

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(s_\pm)=a^{\pm 1}L(s))}

(here 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 s} is a strand 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 s_\pm} is the same strand with a 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 \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 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(\bigcirc)=1} . State and prove the relationship between 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 F} 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 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.

@@ Line 4: / Line 4: @@
 '''Solve the following problems''' and submit them in class by October 29, 2009:
+'''Problem 1.''' Let <math>{\mathfrak g}_1</math> and <math>{\mathfrak g}_2</math> be finite dimensional metrized Lie algebras, let <math>{\mathfrak g}_1\oplus{\mathfrak g}_2</math> denote their direct sum with the obvious "orthogonal" bracket and metric, and let <math>m</math> be the canonical isomorphism  <math>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
-'''Problem 1.'''
+{{Equation*|<math>{\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>\Box:{\mathcal A}(\uparrow)\to{\mathcal A}(\uparrow)\otimes{\mathcal A}(\uparrow)</math> is the co-product and <math>{\mathcal T}_{{\mathfrak g}}</math> denotes the <math>{\mathcal U}({\mathfrak g})</math>-valued "tensor map" on <math>{\mathcal A}</math>. Can you relate this with the first problem of [[AKT-09/HW1|HW1]]?
+'''Problem 2.'''
 # Find a concise algorithm to compute the weight system <math>W_{so}</math> associated with the Lie algebra <math>so(N)</math> in its defining representation.
 # Verify that your algorithm indeed satisfies the <math>4T</math> relation.
-'''Problem 2.''' The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations
+'''Problem 3.''' The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations
 {{Equation*|<math>L(s_\pm)=a^{\pm 1}L(s))</math>}}
@@ Line 17: / Line 21: @@
 and by the initial condition <math>L(\bigcirc)=1</math>. State and prove the relationship between <math>F</math> and <math>W_{so}</math>.
-'''Problem 3.'''
 '''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo|photo page]].

AKT-09/HW2: Difference between revisions

Revision as of 12:14, 19 October 2009

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