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# Video 090910-1

Videography by Karene Chu

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Videography by Karene Chu

**Notes on AKT-090910-1:**
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3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.

# | 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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The arcs of can be coloured in 3 colours s.t.

- Every crossing is either monochromatic or trichromatic.
- All colours appear.

__Terminology__ (for the purpose of this annotation)

**Local rule**: at each crossing, of the 3 arcs involved, either 1 or all 3 colours appear.

**Global rule**: all 3 colours must appear.

(Together with the provision of 3 colours, knots which can be coloured obeying these rules, are called **tricolourable** as defined at 0:10:26 of this hour.)

__Problem/Concern__:
Sometimes, in a local picture, only 2 colours appear on one side of an isotopy move (e.g. R2) whereas all 3 colours appear on the other. One might worry that this could lead to the violation of the *global rule*.

__Solution__:
One can prove that a knot (consisting of only 1 connected piece of material in ), which is coloured obeying the local rule of tricolourability, has at least 2 colours it has all 3 colours. The proof relies on the fact that the same piece of material can change colour (from one colour to a 2nd colour) only by going 'under' a crossing, and whenever a crossing involves 2 colours it must involve a 3rd. (This argument fails, however, for links. Just consider two knots, one red and one blue say, placed side by side.)

**Failed to parse (unknown function\slashoverback): \left\langle\slashoverback\right\rangle=A\left\langle\hsmoothing\right\rangle + B \left\langle\smoothing\right\rangle**

These suffice because for any knot or link diagram we can apply the first relation recursively until we eliminate all the crossings and end up with a disjoint union of unknots. For instance, we get:

**Failed to parse (unknown function\HopfLink): \left\langle \HopfLink \right\rangle = A^2d^2 + 2ABd + B^2d^2**