Unfoldings in knot theory. (English) Zbl 0675.57010

This paper presents a topological analogue of the notion of unfolding of an isolated singularity of a complex hypersurface, which applies to all fibred links. (Here a link is a codimension 2 submanifold of a sphere, whose components need not themselves be spherical.) Operations such as connected sum, Murasugi sum and (usually) cabling can be expressed in terms of unfolding. It is also shown that there are strong restrictions on the topology “near infinity” of a complex hypersurface in \({\mathbb{C}}^ n\) with only finitely many singularities. These results are then used to unify and strengthen work of Abhyankar, Moh, Zaidenberg and Lin on polynomial injections of \({\mathbb{C}}\) into \({\mathbb{C}}^ 2\).
[See also the corrigendum (reviewed below, see Zbl 0675.57011) where it is shown that the definition of “good” polynomial map given in Section 1.2 must be modified, in order that Theorem 6.1 and a modified version of Lemma 7.1 be true.]
Reviewer: J.Hillman


57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
32S05 Local complex singularities
57M25 Knots and links in the \(3\)-sphere (MSC2010)
14E25 Embeddings in algebraic geometry
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory


Zbl 0675.57011
Full Text: DOI EuDML


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