Knots: Mathematics with a Twist

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Overview

Ornaments and icons, symbols of complexity or evil, aesthetically appealing and endlessly useful in everyday ways, knots are also the object of mathematical theory, used to unravel ideas about the topological nature of space. In recent years knot theory has been brought to bear on the study of equations describing weather systems, mathematical models used in physics, and even, with the realization that DNA sometimes is knotted, molecular biology.

This book, written by a mathematician known for his own work on knot theory, is a clear, concise, and engaging introduction to this complicated subject. A guide to the basic ideas and applications of knot theory, Knots takes us from Lord Kelvin's early--and mistaken--idea of using the knot to model the atom, almost a century and a half ago, to the central problem confronting knot theorists today: distinguishing among various knots, classifying them, and finding a straightforward and general way of determining whether two knots--treated as mathematical objects--are equal.

Communicating the excitement of recent ferment in the field, as well as the joys and frustrations of his own work, Alexei Sossinsky reveals how analogy, speculation, coincidence, mistakes, hard work, aesthetics, and intuition figure far more than plain logic or magical inspiration in the process of discovery. His spirited, timely, and lavishly illustrated work shows us the pleasure of mathematics for its own sake as well as the surprising usefulness of its connections to real-world problems in the sciences. It will instruct and delight the expert, the amateur, and the curious alike.

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Editorial Reviews

Science News
The author describes knot theory by chronicling its history. Beginning with Lord Kelvin's ill-conceived idea of using knots as a model for the atom, Sossinsky moves to the connection of knots to braids and then on to the arithmetic of knots. Other topics are the Jones polynomial, which links knot theory to physics, and a clear exposition on Vassilev invariants. Throughout, this book untangles many a snag in the field of mathematics.
Choice

In a charming and spirited discussion of classical and contemporary knot theory, Sossinsky, beginning with Lord Kelvin's (c. 1860) theory of knots as models for atoms...moves through discussions of braids, links, Reidemeister moves, surgery, various knot polynomials (Alexander-Conway, Homfly, Jones), Vassiliev invariants, and concludes with connections between and speculations about knots and physics.
— S. J. Colley

The Guardian
This eminently likeable introduction to knot theory is heavily illustrated with diagrams to help us get our heads around the mind-bending ideas, and Sossinsky delights in breaking off at tangents to relate surprising knot-related facts of the natural world, such as the fish that ties its body in a knot to escape predators, or the topological operations that are performed by an enzyme on DNA.
Daniel Goroff
[A] thought-provoking analysis of why technology has failed to live up to its promises.
Clifford Pickover
In her provocative new book, Victoria Nelson contends that modern civilization has repressed our spiritual instincts.
Ian Stewart
Nelson skillfully manages to thrust the sphere of academic research headlong into popular culture, making this both accessible and erudite...
Choice - S. J. Colley
In a charming and spirited discussion of classical and contemporary knot theory, Sossinsky, beginning with Lord Kelvin's (c. 1860) theory of knots as models for atoms...moves through discussions of braids, links, Reidemeister moves, surgery, various knot polynomials (Alexander-Conway, Homfly, Jones), Vassiliev invariants, and concludes with connections between and speculations about knots and physics.
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Product Details

  • ISBN-13: 9780674013810
  • Publisher: Harvard University Press
  • Publication date: 3/28/2004
  • Pages: 160
  • Product dimensions: 5.00 (w) x 6.92 (h) x 0.48 (d)

Meet the Author

Alexei Sossinsky is Professor of Mathematics, University of Moscow.
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Table of Contents

Preface

1. Atoms and Knots
Lord Kelvin — 1860

2. Braided Knots
Alexander — 1923

3. Planar Diagrams of Knots
Reidemeister — 1928

4. The Arithmetic of Knots
Schubert — 1949

5. Surgery and Invariants
Conway — 1973

6. Jones's Polynomial and Spin Models
Kauffman — 1987

7. Finite-Order Invariants
Vassiliev — 1990

8. Knots and Physics
Xxx? — 2004?

Notes

Works Cited

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