Hyperbolic knot theory

by Purcell, Jessica

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Title Statements

Full title :Hyperbolic knot theory / Jessica S. Purcell.

Summary

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. The book was written to be interactive, with many examples and exercises. Some important results are left to guided exercises. The level is appropriate for graduate students with a basic background in algebraic topology, particularly fundamental groups and covering spaces. Some experience with some differential topology and Riemannian geometry will also be helpful.

Edition Statements

Publication Date:©2020

Publication Statements

Publisher(s):American mathematical Society
Place of Publication:Providence, Rhode Island
Language:eng
Publication Date:©2020

The Physical Object

Description:1 online resource (xviii, 369 pages) : : illustrations (some color).

Edition Identifiers

ISBN:9781470462116, 1470462117
oclc:on1195924383
LC Classification:QA612.2
Dewey Decimal:514/.2242

Decomposition of the Figure-8 Knot -- Calculating in Hyperbolic Space - Geometric Structures on Manifolds -- Hyperbolic Structures and Triangulations -- Discrete Groups and the Thick-Thin Decomposition -- Completion and Dehn Filling -- Twist Knots and Augmented Links -- Essential Surfaces -- Volume and Angle Structures -- Two-Bridge Knots and Links -- Alternating Knots and Links -- The Geometry of Embedded Surfaces -- Estimating Volume -- Ford Domains and Canonical Polyhedra -- Algebraic Sets and the A-Polynomial.

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