Constructing essential laminations in 2-bridge knot surgered 3-manifolds
type: Item
instance of: scholarly article
title: Constructing essential laminations in 2-bridge knot surgered 3-manifolds
zbMATH Open document ID: 0887.57020
author: Ramin Naimi
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: A nontrivial knot that can be drawn with only two relative maxima in the vertical direction is called a 2-bridge knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking, a lamination in a manifold \(M\) is a foliation of \(M\), except that it can have a nonempty open complement in \(M\), and very loosely speaking, the lamination is essential if each leaf of it, \(L\), is incompressible, i.e. inclusion of \(L\) into \(M\) induces an injective homomorphism from \(\pi_1 (L)\) into \(\pi_1 (M)\). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a non-torus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by \(\mathbb{R}^3\), and have infinite fundamental group. So property P is true for non-torus 2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or a counterexample to the Poincaré conjecture. We use general techniques which are not specific to 2-bridge knots to find and explicitly construct these laminations.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 950211
zbMATH Keywords: 3-manifold
zbMATH Keywords: 2-bridge knot
zbMATH Keywords: essential laminations
MaRDI profile type: Publication
full work available at URL: pjm.1997.180.153
OpenAlex ID: W2103057558
DOI: 10.2140/PJM.1997.180.153
description: scientific article; zbMATH DE number 950211