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Algebraic invariants of links. (English) Zbl 1007.57001
Series on Knots and Everything. 32. Singapore: World Scientific. xii, 305 p. (2002).
This beautiful book is a fundamental work on links and the main algebraic invariants arising from covering spaces of link exteriors. Many arguments discussed in the text are not usually considered in standard textbooks on the topic. Among them, there are, for example, the homological complexity of many-variable Laurent polynomial rings, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completions, algebraic closures of link groups, and disc links. We remark that invariants of these types play an essential role in many applications of knot and link theory to other areas of topology, and especially the topology of manifolds.
Chapter 1 is devoted to a basic treatment of knots and links from a geometric point of view, and the main equivalence relations (that is, isotopy, concordance, I-equivalence, and link homotopy) used in classifying them are presented. The algebraic invariants of links arising from those of covering spaces of their exteriors are largely discussed in Chapter 2. They are, for example, homology groups of the link exteriors considered as modules over the group ring of the covering groups, bilinear pairings determined by PoincarĂ© duality, the Blanchfield pairings associated to free abelian covers, the total linking number cover, the localized Blanchfield pairing on the maximal abelian cover, and the signature invariants. In Chapter 3 one can find a detailed description of the main determinantal invariants of modules and chain complexes over a Noetherian ring, namely the elementary ideals, their divisorial hulls, the Reidemeister-Franz torsion, the Steinitz-Fox-Smythe invariant, and bilinear pairings. These invariants are then considered for the homology of abelian covers of link exteriors in Chapters 4 and 5. This completes the first part of the book, named ‘Abelian Covers’.
In the second part of it (called ‘Applications: Special Cases and Symmetries’), the above ideas and algebraic constructions are applied to study special classes of links (as, for examples, knots, links with null Alexander polynomial, 2-component \(\mathbb Z_2\)-boundary links, and 2-component links) and symmetries (as, for example, symmetries of knot types, links with infinitely many semifree periods, knots with free periods, strong symmetries, and equivariant concordance). In particular, relations between symmetries of links and link types with the Alexander invariants are treated in Chapter 8.
In the third part of the book, named ‘Free Covers, Nilpotent Quotients and Completion’, the author describes many algebraic invariants of nonabelian coverings (as free covers, nilpotent quotients, solvable quotients, and completion) and studies their applications to geometric questions of concordance and link homotopy. A final discussion on disc links and string links completes the book.
The author, who is one of the major experts on the topic, must be surely congratulated for this attractive book, written in a careful, very precise and quite readable style. It serves as an excellent self-contained and up-to-date monograph on the algebraic invariants of links, also supplemented by the other fine books of the author [Alexander ideals of links, Lect. Notes Math. 895 (1981; Zbl 0491.57001); 2-knots and their groups, Aust. Math. Soc. Lect. Ser. 5 (1989; Zbl 0669.57008)].
The reader is expected to be familiar with elementary topology and algebra. It is also assumed that the reader knows already some of the standard algebraic techniques of homological algebra. An extensive literature on the subject of this book is listed and quoted in the references. For experts in knot and link theory, the book offers a detailed exposition of many elegant results and applications of the algebraic theory of covering spaces of link exteriors to the theory of links.
Finally, I strongly recommend this beautiful book to anyone interested in the algebraic theory of links and its applications.

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)