This book is a much-expanded new edition of a book first published in 1983 by Princeton University Press, number 30 in their "Mathematical Notes" series. The 1983 "Telegraphic Review" described the book as

Exploration in combinatorics and knot theory, where knots are modelled by a state polynomial. A key combinatorial result gives rise to a duality conjecture, a refinement of the Alexander polynomial, and extension of the Crowell-Murasugi theorem to the class of links.

The focus of the book is very much on knot *diagrams*, the two-dimensional drawings that describe knots. It uses some rather "startling" (Kauffman's own description!) terminology: the knot diagram is called a "universe," with "singularities" (the crossings). Universes can have "states," "black holes," "white holes," and "stars"...

The author has added two new chapters. "Remarks on Formal Knot Theory" is a guide to reading the original book, complementing it and serving as an introduction to further developments in knot theory since the original publication. "New Invariants in the Theory of Knots" is an article that originally appeared in the *American Mathematical Monthly* (March, 1988); it deals mostly with the Jones polynomial and related topics.

The two new chapters amount to almost 100 pages, a significant expansion of the original 167-page notes. Their addition makes this an essentially new book, so that even those who already have a copy of the older edition will want to take a look at this one.

Fernando Q. Gouvêa is the editor of MAA Reviews.