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**Notes on AKT-140106:**
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Course introduction, knots and Reidemeister moves, knot colourings.

# | 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: Self-linking using swaddling. Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres. |

3 | Jan 20 | 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. 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 . |

5 | Feb 3 | 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 . |

6 | Feb 10 | 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. |

R | Feb 17 | Reading Week. |

7 | Feb 24 | 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). |

8 | Mar 3 | 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. |

9 | Mar 10 | 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. |

10 | Mar 17 | Homework Assignment 8 (PDF) Monday: and PBW. Wednesday: The anomaly. Friday: Faddeev-Popov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF). |

11 | Mar 24 | Homework Assignment 9 (PDF) Monday: is a bi-algebra. 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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**Claim:** The number of legal 3-colorings of a knot diagram is always a power of 3.

This is an expansion on the proof given by Przytycki (https://arxiv.org/abs/math/0608172).

We'll show that the set of legal 3-colorings forms a subgroup of , for some r, which suffices to prove the claim. First, label each of the segments of the given diagram 1 through n, and denote a 3-coloring of this diagram by , where each is an element of the cyclic group of order 3 (each element representing a different colour). It is clear that is a subset of . To show it is a subgroup, we'll take , and show that . It suffices to restrict our attention to one crossing in the given diagram, so we can without loss of generality let n = 3.

First, we (sub)claim that a crossing (involving colours is legal if and only if in . Indeed, if the crossing is legal, either it is the trivial crossing in which case their product is clearly 1, or each is distinct, in which case . Conversely, suppose , and suppose . It suffices to show that . This follows by case checking: if , then ; if , then , implying that ; and if , then , implying that . Thus, the subclaim is proven.

As a result, satisfies since both . This implies that , and hence shows that is a subgroup of for n = the number of line segments in the diagram. By Lagrange's theorem, the number of legal 3-colorings (the order of ) is a power of 3.

(Note by User:Leo algknt):

**Using linear Algebra: Idea from class on Wednesday 23 May, 2018**

Let be a knot diagram for the knot with crossings. There are arcs. Let represent the arcs. Now let . Define by

,
so that .

Then, with the above definition, we get a linear equation for each each of the crossings, where . Thus we get a system of linear equation, from which we get a matrix . The nullspace of is the solution to this system of equation and this is exactly the set of all 3-colourings of . This is a vector space of size