The coarse geometry of the Kakimizu complex
The Kakimizu complex of a knot \(K\) is a simplicial complex that records the structure of the set of minimal genus Seifert surfaces for \(K\). Here the surfaces are considered up to ambient isotopy in the complement of \(K\). The definition of the complex mirrors that of the curve complex of a surface: vertices correspond to minimal genus Seifert surfaces, and higher dimensional simplices are given by disjointness. For a hyperbolic knot the Kakimizu complex is finite, whereas for satellite knots it can be infinite and even locally infinite.NEWLINENEWLINEThe coarse geometry of a metric space describes `what it looks like from far away'. More precisely, the aim is to classify spaces up to quasi-isometry. The Kakimizu complex of \(K\) inherits a metric from its simplicial structure, while \(\mathbb{Z}^n\) inherits a metric from \(\mathbb{R}^n\). In this paper it is shown that, for a knot \(K\), there is some \(n\geq 0\) such that the Kakimizu complex of \(K\) is quasi-isometric to \(\mathbb{Z}^n\). The value of \(n\) is not explicitly calculated, although it can be estimated from the proof.NEWLINENEWLINEThe proof makes use of the JSJ tori in the complement of the knot. One method for constructing minimal genus Seifert surfaces is to `wind around' these tori. The authors identify a section of the JSJ decomposition of the knot \(K\), which they call the `core'. Using the JSJ tori associated with the core they construct a group action on the Kakimizu complex, from which they build the quasi-isometry. Another piece of the proof is the classification of the Seifert fibred submanifolds of \(S^3\) by \textit{R. Budney} [Enseign. Math. (2) 52, No. 3--4, 319--359 (2006; Zbl 1114.57004)].
scientific article; zbMATH DE number 6369090
scientific article