Knots: Mathematics with a Twist

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.

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

About the Author :
Alexei Sossinsky is Professor of Mathematics, University of Moscow.

Review :
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. 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. 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. Indeed, knots are trendy and also accessible to recreational mathematicians. A sophisticated high school student might enjoy working out the math in this book, while a full-fledged math student would find it a charming tour of knot theory’s greatest hits… An enjoyable math book and highly recommended. Knots is a spirited, timely, and sound book by a mathematician who knows the technicalities but writes appealingly for the general reader. It’s not often that a book can take you to the frontiers of mathematical research as pleasantly as Knots. Using nothing more complicated than a few simple diagrams, Alexei Sossinsky leads his readers on a gripping journey to the cutting edge of modern topology, with a hint at its deep connections with quantum physics. Knots have fascinated humanity since the dawn of civilization and have permeated virtually all aspects of our lives, from engineering to sailing to knitting. Anyone interested in the beauty and mystery of knots will enjoy this amazing book.