Hyperbolic Knot Theory
Hyperbolic Knot Theory (Graduate Studies in Mathematics)
by Jessica S. Purcell
English | 2020 | ISBN: 1470454998 | 392 Pages | PDF | 33 MB
by Jessica S. Purcell
English | 2020 | ISBN: 1470454998 | 392 Pages | PDF | 33 MB
This book provides an introduction to hyperbolic geometry in dimension three, with motivation and applications arising from knot theory. Hyperbolic geometry was first used as a tool to study knots by Riley and then Thurston in the 1970s. By the 1980s, combining work of Mostow and Prasad with Gordon and Luecke, it was known that a hyperbolic structure on a knot complement in the 3-sphere gives a complete knot invariant. However, it remains a difficult problem to relate the hyperbolic geometry of a knot to other invariants arising from knot theory. In particular, it is difficult to determine hyperbolic geometric information from a knot diagram, which is classically used to describe a knot. This textbook provides background on these problems, and tools to determine hyperbolic information on knots. It also includes results and state-of-the art techniques on hyperbolic geometry and knot theory to date.