×

On maximal fibred submanifolds of a knot exterior. (English) Zbl 0651.57007

Let K be a knot in the 3-sphere \(S^ 3\), E its exterior and \(G=\pi_ 1(E)\) the group of K. For an incompressible spanning surface \(S\subset E\) for K, we define a subgroup \(\mu\) (S) of G by \(\mu (S)=\cap_{k\in {\mathbb{Z}}}m^ k\pi_ 1(S)m^{-k}\) where m is the element of G corresponding to a meridian. We prove that \(\mu\) (S) is an invariant for K and it is realized by a compact surface m(S) on S; in particular \(\mu\) (S) is a finitely generated free group. Furthermore it is shown that there is a fibred submanifold F(S) of W with fibre m(S) which is maximal in a sense, and that the pair (F(S),m(S)) is unique up to ambient isotopy of E.
Reviewer: O.Kakimizu

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Alford, W.R.: Complements of minimal spanning surfaces of knots are not unique. Ann. Math.91, 419-424 (1970) · Zbl 0193.52901
[2] Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math.115, 83-107 (1966) · Zbl 0136.44605
[3] Haken, W.: Some results on surfaces in 3-manifolds. In: Hilton, P.J. (ed.) Studies in modern topology, Vol. 5. New York: Prentice-Hall 1968 · Zbl 0194.24902
[4] Jaco, W.H.: Lectures on 3-manifold topology. Regional conference series in mathematics, No. 43. Providence, R. I.: American Mathematical Society 1980 · Zbl 0433.57001
[5] Jaco, W.H., Shalen, P.B.: Peripheral structure of 3-manifolds. Invent. Math.38, 55-87 (1976) · Zbl 0395.57004
[6] Kakimizu, O.: Cores of non-compact irreducible 3-manifolds. Kobe J. Math.5, 57-66 (1988) · Zbl 0683.57003
[7] Milnor, J.: Singular points of complex hypersurfaces. Annals of Math. Study 61. Princeton, N.J.: Princeton University Press 1968 · Zbl 0184.48405
[8] Schubert, H.: Knoten und Vollringe. Acta Math.90, 131-286 (1953) · Zbl 0051.40403
[9] Scott, P.: Compact submanifolds of 3-manifolds. J. Lond. Math. Soc.7, 246-250 (1973) · Zbl 0266.57001
[10] Simon, J.: Compactifications of covering spaces of compact 3-manifolds. Mich. Math. J.23, 245-256 (1976) · Zbl 0338.57003
[11] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-88 (1988) · Zbl 0157.30603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.