Hyperbolic Knot Theory

This book provides an introduction to hyperbolic geometry in dimension three,with motivation and applications arising from knot theory. It is difficult to relate the hyperbolic geometry of a knot to other invariants, but the textbook provides tools to determine this information. It is written for graduate students with a basic background in algebraic topology and some experience with differential topology and Riemannian geometry.

Format: Paperback / softback
Length: 369 pages
Publication date: 30 November 2020
Publisher: American Mathematical Society

This comprehensive textbook offers an in-depth introduction to hyperbolic geometry in three dimensions, drawing motivation and applications from the field of knot theory. Initially utilized by Riley and Thurston in the 1970s to study knots, hyperbolic geometry has since evolved to become a powerful tool for understanding these intricate structures. By the 1980s, the work of Mostow and Prasad, combined with Gordon and Luecke, established that a hyperbolic structure on a knot complement in the three-sphere provides a complete knot invariant. However, the challenge of relating hyperbolic geometry to other invariants arising from knot theory remains complex. In particular, determining hyperbolic geometric information from a knot diagram, a classical tool for describing knots, remains a formidable task.

To address these challenges, this textbook provides a comprehensive background on the topics at hand. It offers a range of tools and techniques for determining hyperbolic information on knots, making it an invaluable resource for researchers and students alike. Furthermore, the book includes up-to-date results and state-of-the-art techniques in hyperbolic geometry and knot theory, making it a valuable reference for anyone interested in this field.

Designed to be an interactive textbook, the book is filled with examples and exercises to aid in understanding the concepts. While some important results are left to guided exercises, the level of the text is appropriate for graduate students with a solid background in algebraic topology, particularly fundamental groups and covering spaces. Some experience with differential topology and Riemannian geometry will also be beneficial.

In conclusion, this textbook is a must-read for anyone seeking to delve into the world of hyperbolic geometry and knot theory. With its comprehensive coverage, engaging writing style, and wealth of exercises, it provides an excellent foundation for understanding these complex subjects. Whether you are a student, researcher, or professional in the field, this book will be an invaluable resource for your studies.

Weight: 682g
Dimension: 182 x 255 x 24 (mm)
ISBN-13: 9781470454999