In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such cl
asses of Seifert surfaces that admit disjoint representatives. We show that this complex is simply connected.
The Kakimizu complex is usually defined in the context of knots, where it is known to be quasi-Euclidean. We here generalize the definition of the Kakimizu complex to surfaces and 3-manifolds (with or without boundary). Interestingly, in the setting
of surfaces, the complexes and the techniques turn out to replicate those used to study the Torelli group, {it i.e.,} the nonlinear subgroup of the mapping class group. Our main results are that the Kakimizu complexes of a surface are contractible and that they need not be quasi-Euclidean. It follows that there exist (product) $3$-manifolds whose Kakimizu complexes are not quasi-Euclidean.
