BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

The discovery 18 years ago by Vaughan Jones of a powerful new polynomial invariant for knots and links began a revolution in knot theory. Of the nearly 8,000 Mathematical Reviews about knots, more than two thirds have appeared after 1985. Connections between knot theory and other areas of mathematics as well as physics are no longer surprising. Into this heady environment come two new publications, both revisions of important books on knots and links, both surveying knot theory from an earlier, mostly algebraic point of view. Knots, 2nd edition, by Gerhard Burde and Heiner Zieschang, revises and expands the first edition, which was published in 1985. In this book, which has a strongly group-theoretic flavor, links appear, but high-dimensional knots do not. Knots is intended as a textbook, and according to its authors, inspiration was derived from Dale Rolfson’s classic Knots and Links, Publish or Perish Press, published in 1976 and reprinted in 1990 (with corrections). Algebraic Invariants of Links, by Jonathan Hillman, is a greatly revised and expanded version of Hillman’s Alexander Ideals of Links, Springer-Verlag Lecture Notes in Mathematics, Volume 895, published in 1981. Intended more as a reference rather than a textbook, Algebraic Invariants of Links emphasizes links and highdimensional knots. It is more technical than Knots and requires a deeper algebraic background from the reader. Together these two books offer an algebraic view of knot theory that is both panoramic and penetrating. There are several superb surveys of knot theory ([G79], [KW79], [E99], [LO03], [L03], [T85] for example), and it is not our purpose to compete with them. Rather, we intend to describe some of the algebraic ideas that are central to the two books under review. Details about the books themselves will be found in the final section.