The classification of nonsimple algebraic tangles
type: Item
instance of: scholarly article
title: The classification of nonsimple algebraic tangles
zbMATH Open document ID: 0851.57019
DOI: 10.1007/BF01446301
author: Ying-Qing Wu
published in: Mathematische Annalen
publication date: 1996-11-24T00:00:00Z
full work available at URL: https://eudml.org/doc/165411
review text: A tangle \(T\) is a pair of strings \(t_1 \cup t_2\) properly embedded in a 3-ball \(B\). It is called simple if the complement of an open regular neighborhood of the strings is a simple 3-manifold, i.e. a manifold which is irreducible, \(\partial\)-irreducible, atoroidal, and anannular, or equivalently (by Thurston), a manifold which admits a hyperbolic metric with totally geodesic boundary. A tangle is algebraic in the sense of Conway if it can be obtained by summing rational tangles together in various ways. In general it is difficult to determine whether a tangle is simple. The main theorem of this paper will give a complete classification of all nonsimple algebraic tangles. In particular, if \(T\) is a Montesinos tangle \(M(r_1,r_2)\), then it is nonsimple if and only if \(T = M(r_1,1/2n)\) or \(M(1/p,1/q)\). For Montesinos tangle \(T = M(r_1,\dots, r_n)\) with \(n \geq 3\), it is nonsimple if and only if, for some integer \(k\), either \(r_1\) or \(r_n\) is \(1/2k \bmod Z\). We will show that the simplicity of the tangles given by Jaco, Ruberman and Thurston follows immediately from our result.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 861920
zbMATH Keywords: nonsimple algebraic tangles
zbMATH Keywords: Montesinos tangle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 861920