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Week of...

Notes and Links

1

Jan 6

About This Class (PDF). Monday: Course introduction, knots and Reidemeister moves, knot colourings. Tricolourability without Diagrams Wednesday: The Gauss linking number combinatorially and as an integral. Friday: The Schroedinger equation and path integrals. Friday Introduction (the quantum pendulum)

2

Jan 13

Homework Assignment 1. Monday: The Kauffman bracket and the Jones polynomial. Wednesday: Selflinking using swaddling. Friday: EulerLagrange problems, Gaussian integration, volumes of spheres.

3

Jan 20

Homework Assignment 2. Monday: The definition of finitetype and some examples. Wednesday: The selflinking number and framings. Friday: Integrating a polynomial times a Gaussian. Class Photo.

4

Jan 27

Homework Assignment 3. Monday: Chord diagrams and weight systems. Wednesday: Swaddling maps and framings, general configuration space integrals. Friday: Some analysis of $d^{1}$.

5

Feb 3

Homework Assignment 4. Monday: 4T, the Fundamental Theorem and universal finite type invariants. The FultonMacPherson Compactification (PDF). Wednesday: The FultonMacPherson Compactification, Part I. Friday: More on pushforwards, $d^{1}$, and $d^{\ast }$.

6

Feb 10

Homework Assignment 5. Monday: The bracketrise theorem and the invariance principle. Wednesday: The FultonMacPherson Compactification, Part II. Friday: Gauge fixing, the beginning of Feynman diagrams.

R

Feb 17

Reading Week.

7

Feb 24

Monday: A review of Lie algebras. Wednesday: Graph cohomology and $\Omega _{dR}^{\ast }(\Gamma )$. Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF).

8

Mar 3

Homework Assignment 6 (PDF) Monday: Lie algebraic weight systems. Wednesday: Graph cohomology and the construction of $Z_{0}$. Graph Cohomology and Configuration Space Integrals (PDF) Friday: Gauge invariance, ChernSimons, holonomies. Mar 9 is the last day to drop this class.

9

Mar 10

Homework Assignment 7 (PDF) Monday: The $gl(N)$ weight system. Wednesday: The universal property, hidden faces. Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.

10

Mar 17

Homework Assignment 8 (PDF) Monday: $W_{\mathfrak {g}}\colon {\mathcal {A}}(\uparrow )\to {\mathcal {U}}({\mathfrak {g}})$ and PBW. Wednesday: The anomaly. Friday: FaddeevPopov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF).

11

Mar 24

Homework Assignment 9 (PDF) Monday: ${\mathcal {A}}$ is a bialgebra. Wednesday: Understanding and fixing the anomaly. Friday: class cancelled.

12

Mar 31

Monday, Wednesday: class cancelled. Friday: A Monday class: back to expansions.

E

Apr 7

Monday: A Friday class on what we mostly didn't have time to do.

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Dror's Notebook



Here are some thoughts on how to define tricolourability without choosing a diagram.
Another place in which arcs of a diagram come up naturally is in the Wirtinger presentation for the fundamental group of the knot complement. Here is how the presentation is defined: each arc of the knot diagram corresponds with a generator, and each crossing corresponds with a relation between the generators of the incident arcs of the form $xy=yz$ where $y$ is the generator corresponding with the overcrossing.
Now if $\langle S\mid {\text{relations}}\rangle$ is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map $\phi :S\rightarrow \{R,G,B\}$. We'd like to try to extend this to a group homomorphism $\phi :\langle S\mid {\text{relations}}\rangle \rightarrow \langle R,G,B\mid {\text{relations}}\rangle$. This works if target group has the relation $RG=GB$ along with all other relations obtained by permuting $R,G,B$. These relations fix the target group as $D_{2\cdot 3}$.
Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to $D_{2\cdot 3}$. Not every such homomorphism gives a tricolouring; for example, take the trivial homomorphism. I believe that the following is a sufficient condition for a homomorphism $\phi$ to give a tricolouring: for every element $x$ in $\pi _{1}(\mathbb {R} ^{3}\setminus K)$ whose representative as a loop in $\mathbb {R} ^{3}\setminus K$ has odd linking number with $K$, $\phi (x)$ is an order 2 element in $D_{2\cdot 3}$. Hence, we can define tricolourings as certain kinds of homomorphisms from $\pi _{1}(\mathbb {R} ^{3}\setminus K)$ to $D_{2\cdot 3}$ without having to choose a diagram.