Knots.

*(English)*Zbl 0568.57001
De Gruyter Studies in Mathematics, 5. Berlin - New York: Walter de Gruyter. XII, 399 p. DM 49.95 (1985).

When D. Rolfsen’s book ”Knots and links” appeared in 1976 (Zbl 0339.55004) it became the standard work on knot theory, and for many their introduction to geometric topology. Rolfsen viewed knots and links as key examples in 3-manifold topology, and emphasized the technique of surgery, which leads naturally to higher dimensional considerations. The present authors have produced an excellent text which is quite different from that of Rolfsen, despite sharing much of its material. Their book concentrates on classical knots and is somewhat more combinatorial and less geometric in tone.

Such geometrical ideas as are used tend to be those associated with (Seifert) surfaces, such as the theorems of Nielsen, rather than higher dimensional ones related to surgery, concordance etc. The proofs given have been chosen for their simplicity rather than for their generality; however the more general result is usually stated, and the standard results of 3-manifold topology are invoked where appropriate. The arguments characterizing fibred knots and torus knots in terms of properties of their groups are new and are particularly nice examples of achieving simplicity by using special features of knot complements rather than appealing to general theorems about Haken manifolds. As ”Knots” is typeset (rather than photographically reproduced from typescript) it fits much more detail than ”Knots and links” into about the same number of pages.

There are 15 chapters, including ones on fibred knots, torus knots, uniqueness of factorization, braids, branched coverings, Montesinos links, representations of knot groups and the relationship between knot, exterior and group, as well as the more standard material on Seifert surfaces, Alexander polynomials and so on. There are also short appendices on algebra and on the theorems of Papakyriakopolous and Waldhausen, and 30 pages of tables, giving Alexander polynomials, minimal Seifert matrices, symmetries and cyclic periods of knots up to 10 crossings, inter alia. Finally there is a huge bibliography of approximately 1000 items, which ranges from C. F. Gauss (1833) to V. F. R. Jones (1985). This is accompanied by a very useful key with 32 headings, including a number of aspects of knot theory not touched upon in the text.

Such geometrical ideas as are used tend to be those associated with (Seifert) surfaces, such as the theorems of Nielsen, rather than higher dimensional ones related to surgery, concordance etc. The proofs given have been chosen for their simplicity rather than for their generality; however the more general result is usually stated, and the standard results of 3-manifold topology are invoked where appropriate. The arguments characterizing fibred knots and torus knots in terms of properties of their groups are new and are particularly nice examples of achieving simplicity by using special features of knot complements rather than appealing to general theorems about Haken manifolds. As ”Knots” is typeset (rather than photographically reproduced from typescript) it fits much more detail than ”Knots and links” into about the same number of pages.

There are 15 chapters, including ones on fibred knots, torus knots, uniqueness of factorization, braids, branched coverings, Montesinos links, representations of knot groups and the relationship between knot, exterior and group, as well as the more standard material on Seifert surfaces, Alexander polynomials and so on. There are also short appendices on algebra and on the theorems of Papakyriakopolous and Waldhausen, and 30 pages of tables, giving Alexander polynomials, minimal Seifert matrices, symmetries and cyclic periods of knots up to 10 crossings, inter alia. Finally there is a huge bibliography of approximately 1000 items, which ranges from C. F. Gauss (1833) to V. F. R. Jones (1985). This is accompanied by a very useful key with 32 headings, including a number of aspects of knot theory not touched upon in the text.

Reviewer: J.-A.Hillman

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

20F36 | Braid groups; Artin groups |