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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 November 3, 2009:
Problem 1. Let
and
be finite dimensional metrized Lie algebras, let
denote their direct sum with the obvious "orthogonal" bracket and metric, and let
be the canonical isomorphism
. Prove that
,
where
is the co-product and
denotes the
-valued "tensor map" on
. Can you relate this with the first problem of HW1?
Problem 2.
- Find a concise algorithm to compute the weight system
associated with the Lie algebra
in its defining representation.
- Verify that your algorithm indeed satisfies the
relation.
Problem 3. The Kauffman polynomial
(see [Kauffman]) of a knot or link
is
where
is the writhe of
and where
is the regular isotopy invariant defined by the skein relations
(here
is a strand and
is the same strand with a
kink added) and
Failed to parse (unknown function "\backoverslash"): {\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)}
and by the initial condition
. State and prove the relationship between
and
.
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.