- Lecturer
- Meike Akveld
- Coordinator
- Tommaso Goldhirsch
- Lecture
- Mon 15-17, HG D1.1
- Exercise Class
- Thu 17-18, HG D7.1

Tentative list of topics to be covered:

- Historical background
- Informal introduction and outlook
- Official introduction
- Simple knot invariants
- The Jones polynomial
- Alternating knots
- Surfaces (an overview)
- Seifert surfaces
- The Alexander polynomial

Week | Chapter(s) | Topics | Files and Slides |
---|---|---|---|

01 | Chapter 0 and 1 | History of knot theory, informal definition of a knot and knot equivalence, crossing number, unknot, operations on knots (mirror image, reflection, connected sum), alternating knots, the unknotting number | |

02 | Chapter 2 | Definition of a knot, of knot equivalence, wild and tame knots, polygonal knots, Delta moves, regular projections and knot diagrams | A wild knot and a polygonal knot |

03 | Chapter 2 and 3 | Regular projections are generic, a strange upperbound, crossing number, unknotting number, definition of links, linking number | A complicated unknot and a paper by Coward and Lackenby (2010) |

04 | Chapter 3 | 3-colourings and p-colourings | Example of p-colourings for two simple knots |

05 | Chapter 4 | Definition of the Jones polynomial using the bracket polynomial. Definition of states of diagrams. | The original paper of V.R.Jones A nontrivial link with trivial Jones polynomial Two different knots with the same Jones Polynomial |

06 | Chapter 4 | Description of Kauffman bracket in terms of states, axiomatic definition of the Jones polynomial and its uniqueness using the Skein relation and the concept of complexity, mirror images revisited | A table of the Jones Polynomial from Lickorish Different mirror images with the same Jones Polynomial |

07 | Chapter 5 | Alternating knots: Definition of reduced and minimal knot diagrams, Theorems about relation of breath of Kauffman bracket (a knot invariant) and the number of crossings, statement of the first Tait Conjecture. | Spot the non-alternating knot diagrams Example how the states relates to the number of closed loops |

08 | No lecture | Sechseläuten | |

09 | Chapter 5 | Alternating knots: Proof of the first Tait conjecture, definition of the HOMFLY and the Alexander-Conway polynomial and some properties of the latter. | Properties of the Conway polynomial Proof of proposition from Lickorish' book |

10 | Chapter 6 | A crash course on surfaces: Repetition of compact or closed n-manifolds, quotient topology, adding handles, cross handles and crosscaps. Definition of combinatorial surfaces. Curves and arcs on surfaces (1-sides, 2-sides, separating, non-separating) and non-orientable surfaces. | Which surface is this? |

11 | Chapter 6 | Euler characteristic as an invariant of surfaces, surfaces represented by polygons with a gluing pattern. Genus of a surface. Classification theorems for closed surfaces and for surfaces with boundary. | Classification of surfaces |

12 | Chapter 7 | Seifert surfaces: Definition of a Seifert surface, proof of existence, genus of a knot, Theorem about additivity of the genus, prima knots. | Construction of a Seifert surface |

13 | Chapter 8 | Seifert matrices: Definition, Seifert- (or S-) equivalence of matrices, Theorem about equivalent knots and their Seifert matrices | Example for calculating a Seifert matrix Nested circles See Chapter 5 of "Knot Theory & Its Applications" by Kunio Murasugi |

14 | Chapter 8 | The Alexander polynomial: The determinant of a knot, the Alexander polynomial and some of its properties, skein relation, some examples | Some examples of knots with different and equal Jones and Alexander polynomials See Chapter 6 of "Knot Theory & Its Applications" by Kunio Murasugi |

The new exercises will be posted here on Monday. We expect you to look at the problems and prepare questions for the exercise class on Thursday.

Please hand in your solutions by the following Monday in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

exercise sheet | due by | solutions |
---|---|---|

Exercise sheet 1: Informal introduction and outlook | Monday 25.02.2019 | Solutions 1 |

Exercise sheet 2: Formal introduction to knots | Monday 4.03.2019 | Solutions 2 |

Exercise sheet 3: Reidemeister moves and some simple knot invariants | Monday 11.03.2019 | Solutions 3 |

Exercise sheet 4: p-colourings | Monday 18.03.2019 | Solutions 4 |

Exercise sheet 5: the Jones polynomial I | Monday 25.03.2019 | Solutions 5 |

Exercise sheet 6: the Jones polynomial II | Monday 01.04.2019 | Solutions 6 |

Exercise sheet 7: Alternating knots | Monday 15.04.2019 | Solutions 7 |

Exercise sheet 8: More about polynomial invariants | Monday 29.04.2019 | Solutions 8 |

Exercise sheet 9: Crashcourse on surfaces I | Monday 06.05.2019 | Solutions 9 |

Exercise sheet 10: Crashcourse on surfaces II | Monday 13.05.2019 | Solutions 10 |

Exercise sheet 11: Seifert surfaces | Monday 20.05.2019 | Solutions 11 |

Exercise sheet 12: Seifert matrices | Monday 27.05.2019 | Solutions 12 |

Exercise sheet 13: the Alexander polynomial | Monday 03.06.2019 | Solutions 13 |

There will be NO exercise class on the 11th of April. We will have an exercise class during the last week of the semester; more informations will follow.

time | room | assistant | language |
---|---|---|---|

Thu 17-18 | HG D 7.1 | Tommaso Goldhirsch | en |

- J. Roberts,
*Knot Knotes*, unpublished lecture notes, 2015 - Bibliography