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Knot groups. (English) Zbl 0184.48903

Annals of Mathematics Studies. No. 56. Princeton, N. J.: Princeton University Press. vi, 113 p. (1965).
The book is an excellent addition to the literature and is of special interest to mathematicians interested in 3-dimensional topology or combinatorial group theory. The book is well written and contains most of the important results known about knot groups.
Chapters I and II are devoted to introductory and notational matters. In Chapter III combinatorial covering space theory for 3-manifolds is discussed and an algorithmic method for obtaining a presentation for the fundamental group of a 3-manifold is given. This method is dual to the usual one and utilizes the construction of a maximal cave rather than a maximal tree. In Chapter IV the author discusses the commutator subgroup of a knot group and the Alexander invariants of a knot group. Paramount in this chapter is the discussion of the work of the author, Crowell and Rapaport. In Chapter V subgroups of knot groups are discussed and, in particular, the kernel of a homomorphism onto \(\mathbb Z_n\). The reader should be warned to reconstruct it for himself. In Chapter VI representations of knot groups are discussed, and in particular, representations onto symmetric, alternating, and metacyclic groups. In Chapter VII automorphisms are discussed, particularly automorphisms of knot groups whose commutator subgroups are finitely generated (fibered knots). Theorem 7.2.3 is correct, but the proof in the book seems inconclusive. The results of Chapter VIII are new. The product of knots induces a semigroup of knots, and the author considers this semi-group modulo the semi-group of slice knots. In Chapter IX the author considers the characterization problem for knot groups, and discusses necessary, sufficient, and necessary and sufficient conditions for a group to be a knot group. In Chapter X the author discusses the strength of the group in determining (i) the topological type of the complement and (ii) the knot type. The appendix, written by S. Eilenberg is helpful to an understanding of Chapter VIII.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57K10 Knot theory
57K31 Invariants of 3-manifolds (including skein modules, character varieties)
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