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Dror Bar-Natan:
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# Video 090910-2

Videography by Karene Chu

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Videography by Karene Chu

**Notes on AKT-090910-2:**
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R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.

# | Week of... | Videos, Notes, and Links |
---|---|---|

1 | Sep 7 | 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 |

2 | Sep 14 | 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. |

3 | Sep 21 | 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. |

4 | Sep 28 | 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. |

5 | Oct 5 | 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 . |

6 | Oct 12 | 091013: Algebraic properties of vs. algebraic properties of . Thursday's class canceled. |

7 | Oct 19 | 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. |

8 | Oct 26 | 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). |

9 | Nov 2 | 091103: The details of . 091105-1: Three basic problems: genus, unknotting numbers, ribbon knots. 091105-2: The three basic problems and algebraic knot theory. |

10 | Nov 9 | 091110: Tangles and planar algebras, shielding and the generators of KTG. Homework Assignment 3 No Thursday class. |

11 | Nov 16 | Local Khovanov Homology 091119-1: Local Khovanov homology, I. 091119-2: Local Khovanov homology, II. |

12 | Nov 23 | 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. |

13 | Nov 30 | 091201: The relations in KTG. 091203-1: The Existence of the Exponential Function. 091203-2: The Final Exam, Dror's failures. |

F | Dec 7 | The Final Exam on Thu Dec 10, 9-11, Bahen 6183. |

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panel Managed by dbnvp: You can click on many of the blackboard shots / frame grabs to see their full-size versions.

panel Managed by dbnvp: You can click on many of the blackboard shots / frame grabs to see their full-size versions.

Now, the Kauffman bracket becomes a Laurent polynomial in a single variable, .

- Positive crossing:
**Failed to parse (unknown function\overcrossing): \,\overcrossing\,**

- Negative crossing:
**Failed to parse (unknown function\undercrossing): \,\undercrossing\,**

- The
**writhe**of an oriented knot diagram is the sum of the signs of all the crossings in the diagram.

Therefore, and similarly for a negative kink, i.e. the Jones polynomial is invariant under R1.

- 1. How hard is it to compute I, the colouring invariant. (See AKT-09/Tricolourability for a proposed simple procedure.)

- 2. How powerful is the Jones polynomial? (See the next class for the answer.)