Seminar on hyperbolic knot theory

Sommersemester 2023

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

Office hours / further discussion: after the seminar

Announcements:
• 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.

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 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.

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

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