Small knots in some closed Haken 3-manifolds.
type: Item
instance of: scholarly article
title: Small knots in some closed Haken 3-manifolds.
zbMATH Open document ID: 1039.57002
DOI: 10.1016/S0166-8641(03)00161-5
author: Hiroshi Matsuda
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: A closed orientable surface \(F\) embedded in a compact orientable 3-manifold is essential if \(F\) is a 2-sphere not bounding a 3-ball in \(M\). A manifold is called small if there is no closed essential surface embedded in it. In this paper, the author constructs infinitely many new examples of small 3-manifolds \(M\) with boundary a single torus. Under this construction the manifolds \(M\) also satisfy other properties, for instance, (i) there exists a small hyperbolic knot \(K\) in \(M\) (recall that a knot \(K\) in a closed 3-manifold \(M\) is called hyperbolic if \(M-N(K)\) admits a complete hyperbolic structure of finite volume) and (ii) the manifold \(M\) cannot be obtained by Dehn surgery on any knot in \(S^3\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 2037235
zbMATH Keywords: closed essential surface
zbMATH Keywords: small knot
zbMATH Keywords: Haken 3-manifold
reviewed by: Jorge Luis Ramírez Alfonsín
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00161-5
OpenAlex ID: W2038558320
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description: scientific article; zbMATH DE number 2037235