On Odd‐Dimensional Fibred Knots Obtained by Plumbing and Twisting

Plumbing a Hopf band and doing a Stallings twist on the fibre-surface of a given classical fibred link are two powerful ways to produce other such links [6,15]. In fact Harer has shown that every classical fibred link can be constructed from the unknot using these two operations and their inverses. Melvin and Morton have proved using geometric arguments that there exist genus-two fibred classical knots that cannot be obtained by plumbing only (cf. [10]). We consider in this paper the analogue of these two operations for high odd-dimensional knots. For instance Durfee [3] has shown that all knots that arise as the link of an isolated singularity of a complex hypersurface are obtained by plumbing. For this purpose it is necessary to widen the usual definition of a (2k— l)-dimensional knot to include (k — 2)-connected (2k— l)-dimensional differentiable submanifolds N of S. A knot is called spherical if N is homeomorphic to £2fc-i jhg ciassicai concepts of Mink' and 'knot' correspond therefore in high dimensions to those of'knot' and 'spherical knot' respectively. This is the framework adopted in [3]. The use of the A-cobordism theorem is essential in many of the constructions so that we never consider 3-dimensional knots (that is, the case k = 2). We obtain in particular the following results. A simple high-dimensional fibred knot is obtained by plumbing if and only if it admits a unimodular triangular Seifert matrix (Proposition 2.4). Let / be an integer greater than 1 and let K be a (4/— l)-dimensional spherical knot obtained by plumbing with positive definite intersection form /; then /is an orthogonal sum of copies of the form F8 (Corollary 3.4). For every k ^ 3 there exist (2k— l)-dimensional spherical fibred knots of arbitrarily high genus that cannot be obtained by plumbing and twisting (Theorem 5.5). This result has the following classical counterpart. Either there are unimodular Seifert forms that cannot be realised by fibred classical knots or there are fibred knots that cannot be obtained by plumbing and twisting only. Finally we remark that the question whether all high-dimensional simple fibred knots are obtained by plumbing and deplumbing has an interesting formulation in terms of matrices, the answer to which we do not know.