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

Banks, Jessica E.

Algebr. Geom. Topol. 11, No. 3, 1445-1454 (2011).

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)].

Reviewer: Chuichiro Hayashi (Tokyo)

MSC:

57M25

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

Keywords:

links; Kakimizu complex; Seifert surface

Citations:

Zbl 0139.17102; Zbl 0362.55002; Zbl 0774.57006; Zbl 1152.57007

References:

[1]	G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. (1985) · Zbl 0568.57001
[2]	J R Eisner, Knots with infinitely many minimal spanning surfaces, Trans. Amer. Math. Soc. 229 (1977) 329 · Zbl 0362.55002
[3]	W Jaco, U Oertel, An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23 (1984) 195 · Zbl 0545.57003
[4]	O Kakimizu, Doubled knots with infinitely many incompressible spanning surfaces, Bull. London Math. Soc. 23 (1991) 300 · Zbl 0747.57007
[5]	O Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992) 225 · Zbl 0774.57006
[6]	P Przytycki, J Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces · Zbl 1239.57025
[7]	M Sakuma, Minimal genus Seifert surfaces for special arborescent links, Osaka J. Math. 31 (1994) 861 · Zbl 0871.57010
[8]	M Sakuma, K J Shackleton, On the distance between two Seifert surfaces of a knot, Osaka J. Math. 46 (2009) 203 · Zbl 1177.57006
[9]	R T Wilson, Knots with infinitely many incompressible Seifert surfaces, J. Knot Theory Ramifications 17 (2008) 537 · Zbl 1152.57007