Wintersemester 2023/24

Thursdays, 14:15-15:45, in room INF 205 / SR 4.

**Office hours / further discussion:** after the seminar or by appointment

*23.08.2023*: Talks can be given in English or German depending on the audience.*23.08.2023*: There will be a preliminary meeting at the first date where I will give a quick overview on the content and where we can discuss how to organize the seminar and assign first topics if desired. If you are interested please come to the first meeting or contact me via mail.*19.10.2023*: There are still free slots available. If someone is interested in joining the seminar please contact me.

We say that two knots K and K' are

While knot theory looks at the beginning like an easy playground for topologists, it has turned out to be a surprisingly deep topic with many applications in other fields of mathematics. Many of the simplest and most fundamental questions about knots are still unanswered after more than a century, and the questions that have been solved, have often produced very deep and interesting new theories.

In modern knot theory, one does not consider diagrams of knots as in the figure above. Instead, one considers geometric structures on the complements of knots. Since the work of Thurston, it is known that knots fall into three disjoint classes: torus knots, satellite knots, and

The goal of this seminar is to understand and study these results in more detail. To do this, we will follow the excellent book by Jessica Purcell [P] from 2020.

This seminar is intended for students of mathematics and physics (bachelor, teacher students, master, PhD,...) with basic knowledge of Riemannian geometry (curvature, geodesics, hyperbolic space), differential topology (manifolds, group actions), and algebraic topology (fundamental group, covering spaces) for example to the extend of an introductory lecture on differential geometry. The seminar can also be used as preparation (or as a parallel supplement) for writing a thesis or for a small research project in the area of geometry or topology.

19.10.23 | Preliminary meeting and overview [P, Chapter 0] Marc Kegel |
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23.11.23 | The figure-8 knot complement [P, Chapter 1] Cosima Schmitt |
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30.11.23 14:00-15:30 |
Calculations in hyperbolic space [P, Chapter 2] Johannes Manstein |
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7.12.23 |
no seminar |
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14.12.23 | Hyperbolic structures on manifolds and triangulations [P, Chapter 3 and 4] Marc Kegel |

[P] | J. Purcell,
Hyperbolic Knot Theory, 2020, available online on the arXiv:2002.12652. |