Tunnel number one genus one non-simple knots
type: Item
instance of: scholarly article
title: Tunnel number one genus one non-simple knots
zbMATH Open document ID: 0939.57009
author: Hiroshi Goda
author: Masakazu Teragaito
published in: Tokyo Journal of Mathematics
publication date: 2000-07-09T00:00:00Z
review text: A knot in the 3-sphere is said to be tunnel number one if there exists an arc attached to the knot at its endpoints so that the complement of a regular neighborhood of the resulting complex is a genus two handlebody. For a complex \(C\) in \(S^3\), \(N(C)\) means the regular neighborhood of \(C\) in \(S^3\), and \(E(C)\) means the exterior \(S^3- \operatorname{int} N(C)\). Using the work of \textit{K. Morimoto} and \textit{M. Sakuma} [Math. Ann. 289, No. 1, 143-167 (1991; Zbl 0697.57002)] on tunnel number one non-simple knots, the authors classify tunnel number one genus one non-simple knots as follows: Let \(K_0\) be a non-trivial knot \(T(p,q)\) of type \((p,q)\) in \(S^3\) and \(L=K_1 \cup L_2\) a 2-bridge link \(S(\alpha, \beta)\) of type \((\alpha, \beta)\) in \(S^3\) with \(\alpha \geq 4\). Then there is an orientation preserving homeomorphism \(f: E(K_2) \rightarrow N(K_0)\) which takes a meridian \(m_2 \in \partial E(K_2)\) of \(K_2\) to a regular fiber \(h \in \partial N(K_0) = \partial N(K_0)\) of the Seifert fibration of \(E(K_0)\). Denote the knot \(f(K_1) \in N(K_0) \in S^3\) by \(K(\alpha, \beta; p,q)\). Then a tunnel number one genus one non-simple knot \(K\) in \(S^3\) is \(K=K(8m,4m+1;p,q)\) for some \(m \neq 0\).
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1340490
zbMATH Keywords: tunnel number
zbMATH Keywords: classification of knots
reviewed by: Claus Ernst
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270041615
OpenAlex ID: W2060491970
DOI: 10.3836/TJM/1270041615
description: scientific article; zbMATH DE number 1340490