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Hyperbolic Knot Theory
 
Jessica S. Purcell Monash University, Clayton, Victoria, Australia
Hyperbolic Knot Theory
Hyperbolic Knot Theory
Softcover ISBN:  978-1-4704-5499-9
Product Code:  GSM/209
List Price: $98.00
MAA Member Price: $88.20
AMS Member Price: $78.40
Sale Price: $63.70
eBook ISBN:  978-1-4704-6211-6
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:  978-1-4704-5499-9
eBook: ISBN:  978-1-4704-6211-6
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
Hyperbolic Knot Theory
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Hyperbolic Knot Theory
Hyperbolic Knot Theory
Jessica S. Purcell Monash University, Clayton, Victoria, Australia
Softcover ISBN:  978-1-4704-5499-9
Product Code:  GSM/209
List Price: $98.00
MAA Member Price: $88.20
AMS Member Price: $78.40
Sale Price: $63.70
eBook ISBN:  978-1-4704-6211-6
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:  978-1-4704-5499-9
eBook ISBN:  978-1-4704-6211-6
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 Details
     
     
    Graduate Studies in Mathematics
    Volume: 2092020; 369 pp
    MSC: 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 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.

    Readership

    Graduate 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 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
    • Tools, techniques, and families of examples
    • Twist knots and augmented links
    • Essential surfaces
    • Volume and angle structures
    • Two-bridge 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2092020; 369 pp
MSC: 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 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.

Readership

Graduate students interested in hyperbolic geometry and knot theory.

  • Chapters
  • A brief introduction to hyperbolic knots
  • Foundations of hyperbolic structures
  • 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
  • Tools, techniques, and families of examples
  • Twist knots and augmented links
  • Essential surfaces
  • Volume and angle structures
  • Two-bridge 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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