The coarse geometry of the Kakimizu complex
type: Item
instance of: scholarly article
title: The coarse geometry of the Kakimizu complex
zbMATH Open document ID: 1302.57022
published in: Algebraic \& Geometric Topology
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.0530
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 6369090
zbMATH Keywords: knot
zbMATH Keywords: Seifert surface
zbMATH Keywords: Kakimizu complex
zbMATH Keywords: JSJ tori
zbMATH Keywords: quasi-Euclidean
reviewed by: Jessica Banks
MaRDI profile type: Publication
OpenAlex ID: W3105353913
arXiv ID: 1204.0530
cites work: On links with locally infinite Kakimizu complexes
cites work: The Kakimizu complex of a connected sum of links
cites work: JSJ-decompositions of knot and link complements in \(S^3\)
cites work: Q4520856
cites work: Finding disjoint incompressible spanning surfaces for a link
cites work: Classification of the incompressible spanning surfaces for prime knots of 10 or less crossings
cites work: Contractibility of the Kakimizu complex and symmetric Seifert surfaces
cites work: Minimal genus Seifert surfaces for special arborescent links
cites work: On the distance between two Seifert surfaces of a knot
cites work: Finding Disjoint Seifert Surfaces
cites work: KNOTS WITH INFINITELY MANY INCOMPRESSIBLE SEIFERT SURFACES
DOI: 10.2140/AGT.2014.14.2549
author: Jesse Johnson
author: Roberto Pelayo
author: Robin Wilson
review text: 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)].
description: scientific article; zbMATH DE number 6369090