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{{AKT-09/Navigation}} |
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{{AKT-09/Navigation}} |
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'''Solve the following problems''' and submit them in class by October 13, 2006: |
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'''Solve the following problems''' and submit them in class by October 13, 2009: |
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'''Problem 1.''' If <math>f \in {\mathcal V}_n</math> and <math>g \in {\mathcal V}_m</math> then <math>f \cdot g \in {\mathcal V}_{n+m}</math> (as what one would expect by looking at degrees of polynomials) and <math>W_{f \cdot g} = m_\mathbb{Q} \circ (W_f \otimes W_g) \circ \Delta</math> where <math>(W_f \otimes W_g) \circ \Delta: {\mathcal A} \rightarrow \mathbb{Q} \otimes \mathbb{Q}</math> and <math>m_\mathbb{Q}</math> is the multiplication of rationals. (See {{AKT-09/vps|0924-2}}, minute 36:01). |
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'''Problem 1.''' If <math>f \in {\mathcal V}_n</math> and <math>g \in {\mathcal V}_m</math> then <math>f \cdot g \in {\mathcal V}_{n+m}</math> (as what one would expect by looking at degrees of polynomials) and <math>W_{f \cdot g} = m_\mathbb{Q} \circ (W_f \otimes W_g) \circ \Box</math> where <math>(W_f \otimes W_g) \circ \Box: {\mathcal A} \rightarrow \mathbb{Q} \otimes \mathbb{Q}</math> and <math>m_\mathbb{Q}</math> is the multiplication of rationals. (See {{AKT-09/vps|0924-2}}, minute 36:01). |
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'''Problem 2.''' Let <math>\Theta:{\mathcal A}\to{\mathcal A}</math> be the multiplication operator by the 1-chord diagram <math>\theta</math>, and let <math>\partial_\theta=\frac{d}{d\theta}</math> be the adjoint of multiplication by <math>W_\theta</math> on <math>{\mathcal A}^\star</math>, where <math>W_\theta</math> is the obvious dual of <math>\theta</math> in <math>{\mathcal A}^\star</math>. Let <math>P:{\mathcal A}\to{\mathcal A}</math> be defined by |
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'''Problem 2.''' Let <math>\Theta:{\mathcal A}\to{\mathcal A}</math> be the multiplication operator by the 1-chord diagram <math>\theta</math>, and let <math>\partial_\theta=\frac{d}{d\theta}</math> be the adjoint of multiplication by <math>W_\theta</math> on <math>{\mathcal A}^\star</math>, where <math>W_\theta</math> is the obvious dual of <math>\theta</math> in <math>{\mathcal A}^\star</math>. Let <math>P:{\mathcal A}\to{\mathcal A}</math> be defined by |
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# <math>P^2=P</math>. |
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# <math>P^2=P</math>. |
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'''Idea for a good deed.''' Later than October 13, prepare a beautiful TeX writeup (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere. |
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'''Idea for a good deed.''' Later than October 13, prepare a [[AKT-09/Sol1|beautiful TeX writeup]] (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere. |
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'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo|photo page]]. |
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'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo|photo page]]. |
Latest revision as of 13:00, 19 October 2009
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Week of...
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Videos, Notes, and Links
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1
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Sep 7
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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
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2
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Sep 14
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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.
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3
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Sep 21
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090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
090924-1: Some dimensions of , is a commutative algebra, . Class Photo
090924-2: is a co-commutative algebra, the relation with products of invariants, is a bi-algebra.
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4
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Sep 28
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Homework Assignment 1 Homework Assignment 1 Solutions
090929: The Milnor-Moore theorem, primitives, the map .
091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
091001-2: The very basics on Lie algebras.
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5
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Oct 5
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091006: Lie algebraic weight systems, .
091008-1: More on , Lie algebras and the four colour theorem.
091008-2: The "abstract tenssor" approach to weight systems, and PBW, the map .
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6
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Oct 12
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091013: Algebraic properties of vs. algebraic properties of . Thursday's class canceled.
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7
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Oct 19
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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.
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8
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Oct 26
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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).
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9
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Nov 2
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091103: The details of .
091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.
091105-2: The three basic problems and algebraic knot theory.
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10
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Nov 9
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091110: Tangles and planar algebras, shielding and the generators of KTG. Homework Assignment 3 No Thursday class.
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11
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Nov 16
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Local Khovanov Homology
091119-1: Local Khovanov homology, I.
091119-2: Local Khovanov homology, II.
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12
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Nov 23
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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.
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13
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Nov 30
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091201: The relations in KTG.
091203-1: The Existence of the Exponential Function.
091203-2: The Final Exam, Dror's failures.
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F
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Dec 7
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The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
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Register of Good Deeds / To Do List
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![AKT-09-ClassPhoto.jpg](/images/thumb/7/76/AKT-09-ClassPhoto.jpg/310px-AKT-09-ClassPhoto.jpg) Add your name / see who's in!
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Solve the following problems and submit them in class by October 13, 2009:
Problem 1. If
and
then
(as what one would expect by looking at degrees of polynomials) and
where
and
is the multiplication of rationals. (See 090924-2, minute 36:01).
Problem 2. Let
be the multiplication operator by the 1-chord diagram
, and let
be the adjoint of multiplication by
on
, where
is the obvious dual of
in
. Let
be defined by
Verify the following assertions, but submit only your work on assertions 4,5,7,11:
, where
is the identity map and where
for any two operators.
is a degree
operator; that is,
for all
.
satisfies Leibnitz' law:
for any
.
is an algebra morphism:
and
.
satisfies the co-Leibnitz law:
(why does this deserve the name "the co-Leibnitz law"?).
is a co-algebra morphism:
(where
is the co-unit of
) and
.
and hence
, where
is the ideal generated by
in the algebra
.
- If
is defined by
then
for all
.
.
descends to a Hopf algebra morphism
, and if
is the obvious projection, then
is the identity of
. (Recall that
).
.
Idea for a good deed. Later than October 13, prepare a beautiful TeX writeup (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere.
Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.