One may simply say that the classic knot theory (circle in the 3-space) owes its development to visualization. By drawing a generic projection of a knot in the plane one is imagining, from a knot diagram, that he sees the knot in the space, and the best of all, the methods used to study such knots have general validity. More than 70 years ago E. Artin has shown that knotting is not a 3-dimensional phenomenon [Abh. Math. Semin. Univ. Hamb. 4, 174-177 (1925; JFM 51.0450.02)]. The lack of visualization makes higher dimensional knots more untouchable and abstract. So after the circle in the 3-space the next knots are closed surfaces in the 4-space, usually the 2-sphere in 4-space. By further increase of dimension, one has knots of a closed \(n\)-dimensional manifold \(M^n\) in \(\mathbb{R}^{n+2}\), in particular of the \(n\)-sphere in \(\mathbb{R}^{n+2}\). But during the last three decades, the higher dimensional knot theory, in particular 2-knots in 4-space, is developing faster and faster. To the reviewer’s best knowledge, the title under review is the second monograph treating 2-knots. While the first one by J. Hillman [2-knots and their groups (1989; Zbl 0669.57008)] is algebraic in character and does not contain a picture, this one is geometric in tone, and therefore full of pictures. The forthcoming third monograph by A. Ranicki, advertized by Springer Verlag, has not yet been seen by the reviewer.
Surfaces in \(\mathbb{R}^4\) are not easy to visualize but are subject to our mathematical imagination. If we wish to draw something on a sheet of paper, then we must project and do slices. Projection into \(\mathbb{R}^3\) causes singularities, therefore one considers projections in general position in \(\mathbb{R}^3\), so double points, branch points and triple points occur. Such projected surface in \(\mathbb{R}^3\) together with crossing information makes the diagram of a knotted surface. Slicing such diagrams one gets a series of curves which can be drawn on a sheet of paper and which make a movie of the knotted surface. The simplest way to describe what is this monograph about is to quote the authors first sentence of Preface: “The purpose of this book is to develop the diagramatic theory of knotted surfaces in 4-dimensional space in analogy with the classical theory of knotted and linked circles in 3-space”. Since the same authors have already made many contributions in that direction this monograph is their natural outcome. In order to get an idea what analogies are covered, we shall simply list the titles of chapters from which one can have an impression, what is covered in a particular chapter: 1. Diagrams of Knotted Surfaces; 2. Moving Knotted Surfaces; 3. Braid Theory in Dimension Four; 4. Combinatorics of Knotted Surface Diagrams; 5. The Fundamental Group and The Seifert Algorithm; 6. Algebraic Structures Related to Knotted Surface Diagrams.
Each Chapter introduces the new material throughout the classical case, and is illustrated by examples and supplied with exercises. A person without some training in descriptive geometry may have problems in imagining the diagrams. The Bibliography contains around 400 titles, so it is practically complete up to the moment of publishing the book.