Seminar on hyperbolic knot theory

Wintersemester 2023/24

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

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

Announcements:
• 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.

Content: We take a rope and glue together the ends of the rope. The resulting closed rope will in general be knotted and is therefore called knot. Some examples are shown below.

We say that two knots K and K' are isotopic if K can be deformed continuously (without cutting the rope) into K'. The four knots shown above are all isotopic. In the more precise language of differential topology, we say that a knot is the isotopy class of a smooth embedding of the circle into the 3-sphere. A main goal of knot theory is to develop invariants that can distinguish different knots, understand precisely which properties of a knot are captured by its invariants, and how the different invariants are related to each other.

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 hyperbolic knots (that admit a complete finite volume hyperbolic metric on their complement). Since torus knots and satellite knots are well-understood, many questions in knot theory can be reduced to hyperbolic geometry. On the other hand, the extra hyperbolic geometry at hand, allows us to define new powerful knot invariants and perform fast computations. For example the knot with the complicated diagram (of about 300 crosings) shown below is not difficult to handle if we consider the hyperbolic geometry of its complement.

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.

Credit points: The goal of a scientific seminar is the practice of scientific discourse. For this purpose, you will have to work independently on an advanced mathematical topic and discuss it with your fellow students. This is of course only possible if you are present. To receive study points for this seminar, active participation in the seminar is mandatory. In addition, an informative presentation (no longer than 90 minutes) must be given.

Talks:

Literature:
 19.10.23 Preliminary meeting and overview [P, Chapter 0] Marc Kegel 23.11.23 The figure-8 knot complement [P, Chapter 1] Cosima Schmitt 30.11.23 14:00-15:30 Calculations in hyperbolic space [P, Chapter 2] Johannes Manstein 7.12.23 no seminar 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.