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Three-dimensional link theory and invariants of plane curve singularities. (English) Zbl 0628.57002
Annals of Mathematics Studies, No. 110. Princeton, New Jersey: Princeton University Press. VII, 172 p. Cloth: $ 39.50; Paper: $ 13.95 (1985).
Algebraic links, those imbeddings in \(S^ 3\) of compact 1-manifolds which appear as the links of isolated singularities in \({\mathbb{C}}^ 2\) of a curve \(f(x,y)=0\), provide the motivation for this book. The main tool developed here is a view of the Jaco-Shalen-Johannson-Thurston toroidal decomposition of a link exterior in a homology 3-sphere \(\Sigma\) as a “splice decomposition” of the link. From this point of view algebraic links are a subset of the class of links “graph links”) whose splice components are Seifert fibered spaces (i.e. no hyperbolic 3-manifolds appear as splice components).
The authors consider: (1) diagrammatic representations of the splice decomposition and components of graph links, (2) the translation from Puiseux expansions for the plane curve \(f(x,y)=0\) to these splice diagrams, (3) computations via the splice diagram of algebraic invariants of a graph link (the Alexander polynomial and, often, monodromy are computable; additivity results are proved as tools), and (4) an alternate description of graph links in terms of plumbing. Both the splice diagram and the plumbing graph yield criteria for the algebraicity of a link.
The authors list several questions as test problems for their program (pp. 9-10), and one is left wondering about the main unsettled issue. How may the Seifert form of the link be computed from its splice decomposition, or similar data, and does the Seifert form determine the topological type of an algebraic link? The methods of this monograph give access to the Seifert form over the reals in terms of signature invariants, but do not yet solve the full problem - perhaps an extension of these ideas will do the job.
{Chapters: Introduction. I. Foundations; Appendix; Algebraic links. II. Classification. III. Invariants. IV. Examples. V. Relation to plumbing.}

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M25 Knots and links in the \(3\)-sphere (MSC2010)
32S05 Local complex singularities
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