Sommersemester 2023

Freitags, 09:20-10:50, in room 2.006 (RUD 25).

**Office hours / further discussion:** after the seminar

*01.01.2023*: Talks can be given in English or German depending on the audience.*01.01.2023*: There will be a preliminary meeting at 14.02., 11:15-12:45 in room 1.115 (RUD 25) where I will give a quick overview on the content and where we can assign first topics if desired.*21.04.2023*: We have slightly changed the time to 09:20-10:50.

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 book by Jessica Purcell [P] from 2020, which I have not yet read myself, but about which I have heard without exception only positive.

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 my lecture on Differential geometry I. 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.

14.02.23 | Preliminary meeting and overview [P, Chapter 0] 11:15-12:45 in room 1.115 (RUD 25) Marc Kegel |
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21.04.23 | The figure-8 knot complement [P, Chapter 1] Marc Kegel |
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Calculations in hyperbolic space [P, Chapter 2] Recap from your differential topology I lecture in self-study |
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28.04.23 | Geometric structures on manifolds [P, Chapter 3] Wendy Zhangwennan |
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05.05.23 | Hyperbolic structures and triangulations [P, Chapter 4] Chun-Sheng Hsueh |
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12.05.23 | Discrete groups and the thick-thin decomposition [P, Chapter 5] Ben Eltschig |
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19.05.23 | Hyperbolic Dehn filling [P, Chapter 6] Leo Mousseau |
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26.05.23 | no seminar |
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02.06.23 | Twist knots and augmented links [P, Chapter 7] Farid A. Azar Leon |
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09.06.23 | Essential surfaces [P, Chapter 8] Leonard Leass |
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16.06.23 | Volume and angle structures [P, Chapter 9] Chun-Sheng Hsueh |
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23.06.23 | Two-bridge knots and links [P, Chapter 10] Dongyu Lin |
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30.06.23 | Alternating knots and links [P, Chapter 11] Luis Kristic |
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07.07.23 | The geometry of embedded surfaces [P, Chapter 12] Frank Selensky |
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14.07.23 | Estimating the volume [P, Chapter 13] Arthur Berns |
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21.07.23 | Algebraic sets and the A-polynomial [P, Chapter 15] Jonas Miehling |

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