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Knot theory and its applications. Transl. from the Japanese by Bohdan Kurpita. (English) Zbl 0864.57001
Basel: Birkhäuser. 341 p. (1996).
More than most other fields of mathematics, knot theory has a strong experimental flavor dealing with concrete objects in our ambient 3-dimensional space. Various of its basic problems, being subject to our “every-day experience”, can be explained in an intuitive way without using sophisticated mathematical formalisms. Thus it can serve also to understand how rigorous mathematical concepts and proofs can be developed from such intuitive notions. The present book, an extended version of the original Japanese one from 1993, develops knot theory from such an intuitive geometric-combinatorial point of view, avoiding completely more advanced concepts and techniques from algebraic topology as fundamental group/combinatorial group theory, homology theory, theory of 3-manifolds etc. It is intended also for readers without a considerable background in mathematics, and in fact particular attention is given to connections and applications to other natural sciences. Thus the emphasis is on a lucid and intuitive exposition accessible to a broader audience, that is on a careful explanation and motivation of the main concepts and problems of knot theory using many examples, figures, exercises, historical remarks, comments on important unresolved and resolved problems. Nevertheless the book is interesting also from a purely mathematical point of view as it covers a relevant amount of classical knot theory, giving rigorous proofs of various central results (though mathematical purity is not aimed at), as well as discussing some of the new and exciting developments starting with the discovery of the Jones polynomial (“The Jones revolution” is the title of one of the 15 chapters), its connections with statistical mechanics and, finally, Vassiliev invariants. I think the book found a good way between and avoiding a purely descriptive accumulation of facts and concepts and, on the other hand, heavy mathematical formalisms and technicalities often appreciated only by a quite restricted group of specialists. This was made possible by concentrating on the strong combinatorial aspect of the subject, in particular of the above-mentioned new developments.
Roughly, one can distinguish three parts of the book. The first four chapters give a general introduction to the main concepts and problems of knot theory discussing in particular classical knot invariants as minimal crossing number, bridge number, linking number etc. The equivalence of knots is defined in a combinatorial way, and it is shown that any two projections of equivalent knots are related by the Reidemeister moves (which are of course at the basis of every combinatorial development of knot theory). This first part is mainly descriptive and experimental and sets the combinatorial basis for the later chapters. The second part, chapters 5 to 10, is the most “classical” one; it develops the Alexander-Conway polynomial and its basic properties, starting from the concept of a Seifert surface for a knot, associated Seifert matrices and their equivalence under the Reidemeister moves. This is applied to torus knots, their classification and the computation of invariants as signature and genus. Also, tangles and 2-bridge knots, braids, Dehn-surgery and cyclic branched coverings are discussed. The main points of the final part, chapters 11 to 15, are the Jones polynomial and its generalizations, connections with statistical mechanics and Vassiliev invariants. It contains also a discussion of possible applications to biology by using the language of tangles (“DNA and knots”), a discussion of the chromatic polynomial for graphs (which, when embedded in 3-space, are natural generalizations of knots and links) and some application to chemistry (chirality of spatial graphs/molecules).
The book appears in a favourite moment. After a period of relative isolation where knot theory served mainly both as a source and target of concepts and ideas from algebraic topology and combinatorial group theory, the recent developments initiated by Jones and Witten brought it to the center of some of the most important developments in mathematics and mathematical physics of the last decade. The book, written in a stimulating and original style, will serve as a first approach to this interesting field, for readers with various backgrounds in mathematics, physics etc. It is the first text developing recent topics as the Jones polynomial and Vassiliev invariants on a level accessible also for non-specialists in the field.

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
92C40 Biochemistry, molecular biology
05C15 Coloring of graphs and hypergraphs