On links with locally infinite Kakimizu complexes. (English) Zbl 1226.57005

H. Schubert and K. Soltsien showed that a simple knot has only finitely many isotopy classes of minimal genus Seifert surfaces [Abh. Math. Semin. Univ. Hamb. 27, 116–123 (1964; Zbl 0139.17102)]. On the other hand, J. R. Eisner showed that there exists a composite knot with infinitely many classes of minimal genus Seifert surfaces [Trans. Am. Math. Soc. 229, 329–349 (1977; Zbl 0362.55002)]. Let \(L\) be a link such that every minimal genus Seifert surface for \(L\) is connected. Then the Kakimizu complex for \(L\) is a simplicial complex such that its vertices are the isotopy classes of minimal genus Seifert surfaces of \(L\), and such that \(n+1\) vertices span an \(n\)-simplex exactly when they can be realised disjointly in the exterior of \(L\) [O. Kakimizu, Hiroshima Math. J. 22, No. 2, 225–236 (1992; Zbl 0774.57006)]. P. Przytycki and J. Schultens generalized the Kakimizu complex to more general \(3\)-manifolds, and raised the question whether the complex can be locally infinite [“Contractibility of the Kakimizu complex and symmetric Seifert surfaces”, arXiv:1004.4168].
The paper under review answers this question affirmatively, giving an example of a satellite knot \(K\) which has a minimal genus Seifert surface \(R\) and infinitely many minimal genus Seifert surfaces \(R_0, R_1, R_2, \dots\) which are disjoint from \(R\). The knot \(K\) is a Whitehead double of the trefoil knot, and \(R\) a surface of genus \(1\) contained in a solid torus \(V\) whose core forms the trefoil knot. There is an annulus \(A\) connecting \(R\) and the boundary torus \(\partial V\). We assume that the Whitehead double is taken so that the circle \(A \cap \partial V\) is parallel to the boundary circles of an essential annulus \(S_1\) in the trefoil knot exterior. To obtain \(R_0\), we perform a surgery on \(R\) along \(A\), and then take a union of the resulting surface and \(S_1\). Then \(R_n\) is obtained from \(R_0\) by twisting along the torus \(\partial V\).
Moreover, it is also shown that a link \(L\) is a satellite of either a torus knot, a cable knot or a connected sum, with winding number \(0\), if every minimal genus Seifert surface for \(L\) is connected and the Kakimizu complex of \(L\) is locally infinite. The proof is based on a special presentation of a Seifert surface as a Haken sum of normal surfaces given in [R. T. Wilson, J. Knot Theory Ramifications 17, No. 5, 537–551 (2008; Zbl 1152.57007)].


57M25 Knots and links in the \(3\)-sphere (MSC2010)
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