The 39th Annual
(Online) Workshop in

The featured speaker is Jessica Purcell of Monash University, who will give a series of three onehour lectures on using hyperbolic geometry as a tool to study 3manifolds and knot theory. An abstract can be found below.
The workshop will also feature contributed talks of 20 minutes. Contributed talks need not be directly related to the topic of the principal lectures. If you wish to apply to give a contributed talk please submit a title and abstract on the registration form or send directly to Greg Friedman (send email). Talks will be selected in a way that provides a balanced collection of topics and respects the historical traditions of the workshop. Earlier responses may be given some preference. Applicants will be notified whether their talk has been accepted by May 23.
For over four decades, hyperbolic geometry has been used as a tool to study and classify 3manifolds. Hyperbolic 3manifolds are known to satisfy important rigidity results, so that hyperbolic structures are unique or wellbehaved. Thus hyperbolic invariants, such as volume, areas of embedded surfaces, and lengths of short geodesics, become topological 3manifold invariants. However, many of the 3manifolds that arise in geometric topology, such as examples from knot theory, dynamics, or piecewise linear topology, are constructed in ways that do not obviously reflect their geometry. Given a 3manifold with a combinatorial or topological description, it can often be difficult to relate that description to hyperbolic geometry of the manifold. In my talks, I will survey some of the ways of relating hyperbolic geometry to combinatorics and topology in dimension three, and some applications.
Day 1: Hyperbolic knots and alternating knots
Hyperbolic geometry has been used since around the mid1970s to study knot theory, but it can be difficult to relate geometry of knots to a diagram of a knot. However, many results from the 1980s and beyond show that alternating knots have diagrams that seem to more closely reflect their geometric properties than other general knots. In the first part, I’ll introduce hyperbolic geometry and knots, and survey some classical results. In the second part, I’ll describe results specific to alternating knots, and explain some of the tools that are useful to prove results in the alternating case. In the last part of the talk, I’ll explain new methods to extend these tools, to extract geometry from much larger families of knots, discussing some of my recent joint work with Josh Howie and with Effie Kalfagianni.
Day 2: Fully augmented links and circle packings
Fully augmented links form a family of hyperbolic links that are a playground for handson hyperbolic geometry. In the first part of the talk, I’ll define the links and show how to determine their hyperbolic geometry explicitly. I will show how such links are determined by circle packings of the sphere, and vice versa. The theory of circle packings extends beyond spheres. I’ll then explain new work, building fully augmented links on higher genus surfaces. In the last part of the talk, I’ll give some applications to geometric convergence of knots and links. The latter part will describe recent joint work with Urs Fuchs and John Stewart.
Day 3: Knots in infinite volume 3manifolds
Classically, knots have been studied in the 3sphere. However, many examples of knots arising in applications lie in broader classes of 3manifolds. These include virtual knots, which lie within thickened surfaces, and periodic orbits of dynamical systems. Not all such manifolds have finite hyperbolic volume. However, the last two decades have brought new tools to bear on hyperbolic geometry of infinite volume 3manifolds. In the first part of the talk, I’ll survey some of those results. I’ll then discuss techniques to mix infinitevolume tools with finitevolume ones, to give further geometric information on knots in general 3manifolds. This includes a 6Theorem for tame 3manifolds, and additional results controlling geometry changes under Dehn filling. The latter part of the talk is joint work with David Futer and Saul Schleimer.
Financial support for the workshop series is provided by a grant from the National Science Foundation (DMS1764311).
Fredric Ancel, University of WisconsinMilwaukee
Greg Friedman, Texas Christian University
Craig Guilbault, University of WisconsinMilwaukee
Molly Moran, Colorado College
Nathan Sunukjian, Calvin College
Eric Swenson, Brigham Young University
Frederick Tinsley, Colorado College
Gerard Venema, Calvin College
Please contact Greg Friedman (send email) if you have questions about the workshop or comments on this web site.