This is the fourth edition of a book that investigates the relationship between knot theory and theoretical physics. It has three distinct parts. The first is a “short course” (of more than 300 pages) on knots and physics that introduces key ideas and examples. The second part is a set of relatively independent excursions into special topics related in one way or another to knot theory and physics. The third is an extended appendix with reprints of six articles by the author.

Kauffman is a mathematician who works in the netherworld between mathematics and theoretical physics. He proves things as a mathematician, but he makes intuitive and sometime mysterious connections between mathematics and physics as a theoretical physicist. Kauffman says: “Acts of abstraction, changes of mathematical mood, shifts of viewpoint are the rule rather than the exception… The course is a rapid guided ascent straight up the side of a cliff!”

The book begins innocently enough with a discussion of actual physical knots: square knot, clove hitch, and bowline. Kauffman then moves on quickly to knot diagrams, the ambient isotopy of knots, and the Reidemeister moves that preserve that isotopy.

The bracket polynomial associated with a knot (related by change of variable to the better known Jones polynomial in one variable) provides a connection to physics. A state summation expression for the bracket polynomial expresses it as a generalized partition function and hence connects it to statistical physics. Another key connection derives from a notational convention that associates abstract (diagrammatic) tensors with knots and enables a generalized multiplication that looks like concatenating components of a knot. This allows one to view the interaction of two particles, for example, as an abstract tensor, consider it as the component of a knot and associate a scattering matrix with it.

From here on we’re headed into the deep: from polynomial invariants of knots and links to Witten’s functional integral formulation of knot and link invariants to topological quantum field theory. While the mathematics here is not inherently difficult, Kauffman presents essentially no background material on the physics. Consequently much of this material — and certainly its physical importance — will be hard to penetrate for many readers.

The second part of the book is a rambling tour of a collection of topics connected in some manner to knots and physics, mathematics and even some philosophy. This part has the best discussion I have seen of what the author calls Dirac’s “belt trick”, a physical manifestation of a loop in SO(3) that is not contractible but whose square is contractible.

This fourth edition includes new material on virtual knot theory and Khovanov homology. Kauffman acknowledges that the book has reached a size where newer editions will be impractical, especially with a flurry of new related work on quantum information theory and loop quantum gravity.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.