Genus one 1-bridge knots as viewed from the curve complex
type: Item
instance of: scholarly article
title: Genus one 1-bridge knots as viewed from the curve complex
zbMATH Open document ID: 1056.57008
author: Toshio Saito
published in: Osaka Journal of Mathematics
publication date: 2004-10-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153493520
review text: In the paper ``3-manifolds as viewed from the curve complex'' [Topology 40, No. 3, 631--657 (2001; Zbl 0985.57014)], \textit{J. Hempel} studied Heegaard splittings of closed 3-manifolds by using the curve complex associated to Heegaard surfaces. In particular, Hempel showed that a Haken manifold is hyperbolic if a Heegaard splitting of the manifold has distance \(\geq 3\). In this paper, a topological characterization of the knots admitting \((1, 1)\)-splittings of distance \(\leq 2\) is given. This characterization implies, as a corollary, that a \((1,1)\)-knot is hyperbolic if and only if has a \((1,1)\)-splitting of distance \(\geq 3\). It is also proved that there are \((1,1)\)-splittings with arbitrarily high distance.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 2111464
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: knot
reviewed by: Jorge Luis Ramírez Alfonsín
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2111464