||Notes and Links
||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)
||Homework Assignment 1. |
Monday: The Kauffman bracket and the Jones polynomial.
Wednesday: Self-linking using swaddling.
Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
||Homework Assignment 2. |
Monday: The definition of finite-type and some examples.
Wednesday: The self-linking number and framings.
Friday: Integrating a polynomial times a Gaussian.
||Homework Assignment 3. |
Monday: Chord diagrams and weight systems.
Wednesday: Swaddling maps and framings, general configuration space integrals.
Friday: Some analysis of .
||Homework Assignment 4. |
Monday: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
Wednesday: The Fulton-MacPherson Compactification, Part I.
Friday: More on pushforwards, , and .
||Homework Assignment 5. |
Monday: The bracket-rise theorem and the invariance principle.
Wednesday: The Fulton-MacPherson Compactification, Part II.
Friday: Gauge fixing, the beginning of Feynman diagrams.
|| Monday: A review of Lie algebras. |
Wednesday: Graph cohomology and .
Friday: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
||Homework Assignment 6 (PDF) |
Monday: Lie algebraic weight systems.
Wednesday: Graph cohomology and the construction of .
Graph Cohomology and Configuration Space Integrals (PDF)
Friday: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
||Homework Assignment 7 (PDF) |
Monday: The weight system.
Wednesday: The universal property, hidden faces.
Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.
||Homework Assignment 8 (PDF) |
Monday: and PBW.
Wednesday: The anomaly.
Friday: Faddeev-Popov, part I.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
||Homework Assignment 9 (PDF) |
Monday: is a bi-algebra.
Wednesday: Understanding and fixing the anomaly.
Friday: class cancelled.
||Monday, Wednesday: class cancelled. |
Friday: A Monday class: back to expansions.
|| Monday: A Friday class on what we mostly didn't have time to do.
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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 where is the generator corresponding with the overcrossing.
Now if is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map . We'd like to try to extend this to a group homomorphism . This works if target group has the relation along with all other relations obtained by permuting . These relations fix the target group as .
Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to . 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 to give a tricolouring: for every element in whose representative as a loop in has odd linking number with , is an order 2 element in . Hence, we can define tricolourings as certain kinds of homomorphisms from to without having to choose a diagram.