Sutured manifolds and generalized Thurston norms
type: Item
instance of: scholarly article
title: Sutured manifolds and generalized Thurston norms
zbMATH Open document ID: 0673.57015
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: Recently, \textit{D. Gabai} has developed the theory of codimension one foliations on sutured 3-manifolds [J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013); ibid. 26, 461-478 and 479-536 (1987; Zbl 0627.57012 and Zbl 0639.57008)]. Using this theory, he has answered positively the Poenaru conjecture, the Property R conjecture, the superadditivity of knot genus under band connected sum, the Property P conjecture for sattelite knots. Eventually it became clear that the theory of sutured 3- manifolds could be developed from a different combinatorial perspective, without foliations. The author gives an account of these developments. The absence of foliations simplifies the proofs. A central role is played by a generalization of the Thurston norm, the so-called \(\beta\)-norm, where \(\beta\) is a properly embedded 1-dimensional complex inside a 3- manifold.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4102087
zbMATH Keywords: beta norm
zbMATH Keywords: Property R
zbMATH Keywords: knot
zbMATH Keywords: Property P
zbMATH Keywords: sutured 3-manifolds
zbMATH Keywords: Thurston norm
author: Martin Scharlemann
reviewed by: Sergei Vladimirovich Matveev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214443063
OpenAlex ID: W1491514327
Wikidata QID: Q115182663
DOI: 10.4310/JDG/1214443063
Recommended article: The sutured Thurston norm
Recommended article: Taut sutured manifolds and twisted homology
Recommended article: Sutured manifold hierarchies, essential laminations, and Dehn surgery
Recommended article: The Thurston norm, fibered manifolds and twisted Alexander polynomials
Recommended article: The dual Thurston norm and the geometry of closed 3-manifolds
Recommended article: The \(L^2\)-torsion function and the Thurston norm of 3-manifolds
Recommended article: Thurston norms of tunnel number-one manifolds
Recommended article: Band-taut sutured manifolds
Recommended article: Thurston Norm and Taut Branched Surfaces
Recommended article: Sutured manifolds and polynomial invariants from higher rank bundles
description: scientific article; zbMATH DE number 4102087