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Knots: Mathematics with a Twist

Alexei Sossinsky
Harvard University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Philip Straffin
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What a beautiful little book! Of course, knot theory is an inherently attractive subject: the objects of study are concrete and physically appealing, the ideas are clever and beautiful and many of them can be understood without extensive mathematical background, there is an abundance of open problems, and there are important modern applications both in mathematics and in biology, chemistry and physics. Still, it takes skill to start at the beginning and steer a clear course through a well-chosen sample of those beautiful ideas to areas of current research — in 126 pages. Sossinsky, a knot theorist from the University of Moscow, has done this with taste, verve and clarity.

The book moves historically, but with free digressions. Each chapter is built around one mathematician's crucial insight:

  1. Lord Kelvin (1860) has the strange idea that atoms might be knots in the ether (but is it any stranger than string theory?), motivating Tait and Little to classify knots with up to ten crossings. Digressions on wild knots and knotted horned spheres.

  2. Alexander (1923) elucidates the relation between knots and braids. This chapter contains an elementary presentation of Vogel's algorithm for turning any knot into a braid.

  3. Reidemeister (1928) shows that all knot deformations can be captured by three types of moves in the knot's planar projection.

  4. Schubert (1949) shows that all knots factor uniquely into prime knots. A wonderful, illustrated digression on how the slime eel knots itself to escape predators.

  5. Conway (1973) shows how to use cutting and pasting on knots ("skein relations") to define polynomial invariants. A digression on how enzymes called topoisomerases already knew how to do Conway's operations on strands of DNA.

  6. Kauffman (1987) shows how to use skein relations to derive the revolutionary new Jones polynomial, which originally came from Jones' work on von Neumann algebras and statistical models in physics. A digression on finding Kauffman's operation on Celtic megaliths.

  7. Vassiliev (1990) extends invariants to singular knots to classify numerical knot invariants.

  8. Xxx ?(2004?), a mathematician or physicist not yet known, finally clarifies intriguing but not understood structural similarities between knot theory and fundamental ideas in physics.

Sossinsky's lively and often personal writing has been transparently translated by Giselle Weiss, and Harvard University Press has made the book look as appealing as its text. However, HUP had more trouble with the mathematics, and there are more mathematical misprints than one would like to see. For instance, in knot diagrams it matters when one strand goes over rather than under another (indeed, that is the idea behind Vassiliev invariants!), and you might enjoy finding the crossing errors in diagrams on pages xi, 88 and 92. Minus signs and exponents always matter in mathematics, and enough errors accumulate on page 88 to require the reader to do an independent calculation of the Jones polynomial for the trefoil knot. Perhaps we should think of these as exercises for the reader...

Although Knots expects a mathematically active reader, it is meant to be an interest-piquing and thought-provoking overview of elegant ideas in knot theory, rather than a text. If it succeeds in its goals for you or your students, you can learn more by going to Charles Livingston's Knot Theory or Colin Adams' The Knot Book, both of which share Sossinsky's expository taste and accessibility, but include more mathematical completeness and detail.

Philip Straffin ( is Professor of Mathematics at Beloit College. As a graduate student he had the pleasure of studying knot theory with Ray Lickorish at Cambridge University, before he strayed into algebraic topology at Berkeley and then more distant areas of mathematics at Beloit. He regularly teaches an undergraduate topology course which includes knot theory and a concrete geometric treatment of surfaces and three-manifolds.