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On even-dimensional fibred knots obtained by plumbing. (English) Zbl 0603.57010

Plumbing is a geometric construction which to any two fibred 2q-knots associates another such knot. A free 2q-knot is one possessing a (q-1)- connected Seifert surface V such that \(H_ q(V)\) is free abelian. The author begins by proving a classification theorem for free fibred 2q- knots, \(q\geq 4\), in terms of equivalence classes of certain associated matrices. This is a generalization of a result of S. Kojima, in that the knots considered are not necessarily spherical but may be embeddings of (q-2)-connected 2q-manifolds in \(S^{2q+2}\). It is shown that there are exactly four such knots of minimal rank, which the author calls Hopf knots.
A 2q-knot is said to be obtained by plumbing if it is obtained from the trivial knot by successively plumbing with Hopf knots. A necessary and sufficient condition for a free fibred knot to be obtained by plumbing is given in terms of its associated matrices, and examples are given of such knots which cannot be obtained by plumbing.
Finally, these concepts are related to the construction known as spinning; in particular, it is shown that the spin of an odd-dimensional knot which is obtained by plumbing (these were dealt with in an earlier paper of the author) is also obtained by plumbing.
Reviewer: Ch.Kearton

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57Q35 Embeddings and immersions in PL-topology
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