© | Dror Bar-Natan: Classes: 2013-14: AKT: | < > |

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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: 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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An invariant $v: \mathcal{VK} \rightarrow A$ (where $\mathcal{VK}$ is the space of virtual knots and $A$ is some abelian group) of virtual knots is said to be of type $n$ if it vanishes on virtual knots with $n+1$ semi-virtual crossings. By virtue of equation (3) above, it can be shown that type $n$ virtual knot invariants, when restricted to honest knots, are knot invariants of type at least n. Namely, a type n virtual knot invariant evaluated on $n+1$ double points is a sum of evaluations of $v$ on $n+1$ semi-virtual crossings (by equation 3), and is therefore equal to 0.