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Softcover ISBN:  9781470454999 
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Softcover ISBN:  9781470454999 
Product Code:  GSM/209 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $78.40 
Sale Price:  $63.70 
eBook ISBN:  9781470462116 
Product Code:  GSM/209.E 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $78.40 
Sale Price:  $63.70 
Softcover ISBN:  9781470454999 
eBook ISBN:  9781470462116 
Product Code:  GSM/209.B 
List Price:  $196.00$147.00 
MAA Member Price:  $176.40$132.30 
AMS Member Price:  $156.80$117.60 
Sale Price:  $127.40$95.55 

Book DetailsGraduate Studies in MathematicsVolume: 209; 2020; 369 ppMSC: Primary 57; Secondary 30;
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 3sphere 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 stateofthe 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.ReadershipGraduate students interested in hyperbolic geometry and knot theory.

Table of Contents

Chapters

A brief introduction to hyperbolic knots

Foundations of hyperbolic structures

Decomposition of the figure8 knot

Calculating in hyperbolic space

Geometric structures on manifolds

Hyperbolic structures and triangulations

Discrete groups and the thickthin decomposition

Completion and Dehn filling

Tools, techniques, and families of examples

Twist knots and augmented links

Essential surfaces

Volume and angle structures

Twobridge knots and links

Alternating knots and links

The geometry of embedded susrfaces

Hyperbolic knot invariants

Estimating volume

Ford domains and canonical polyhedra

Algebraic sets and the $A$polynomial


Additional Material

Reviews

There are many existing books on hyperbolic geometry and on knot theory taken separately, but, to my knowledge, this is the first that substantially focuses on the two fields together. The combination benefits each of the constituents. This book will be useful both as an introduction and as a reference for those interested in either (or both!) topics.
Henry Segerman, Oklahoma State University


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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 3sphere 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 stateofthe 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.
Graduate students interested in hyperbolic geometry and knot theory.

Chapters

A brief introduction to hyperbolic knots

Foundations of hyperbolic structures

Decomposition of the figure8 knot

Calculating in hyperbolic space

Geometric structures on manifolds

Hyperbolic structures and triangulations

Discrete groups and the thickthin decomposition

Completion and Dehn filling

Tools, techniques, and families of examples

Twist knots and augmented links

Essential surfaces

Volume and angle structures

Twobridge knots and links

Alternating knots and links

The geometry of embedded susrfaces

Hyperbolic knot invariants

Estimating volume

Ford domains and canonical polyhedra

Algebraic sets and the $A$polynomial

There are many existing books on hyperbolic geometry and on knot theory taken separately, but, to my knowledge, this is the first that substantially focuses on the two fields together. The combination benefits each of the constituents. This book will be useful both as an introduction and as a reference for those interested in either (or both!) topics.
Henry Segerman, Oklahoma State University