Dehn-surgery along a torus T 2-knot
type: Item
instance of: scholarly article
title: Dehn-surgery along a torus T 2-knot
zbMATH Open document ID: 0629.57009
author: Zyun'iti Iwase
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: A \(T^ 2\)-knot means an embedded 2-torus. The author defines a torus \(T^ 2\)-knot, a generalization of a classical torus knot, as an incompressible embedded 2-torus in the boundary of a tubular neighborhood of an unknotted 2-torus in the 4-sphere, where an unknotted 2-torus means an embedded torus which bounds \(S^ 1\times D^ 2\) in \(S^ 4\). A torus \(T^ 2\)-knot turns out to be an untwisted or twisted spin of a classical torus knot. Dehn surgery means an operation of cutting off the tubular neighborhood of a knot and pasting it back via a diffeomorphism between the boundaries. The author shows that a 4-manifold obtained by Dehn surgery along a torus \(T^ 2\)-knot is (i) the total space of a good torus fibration over \(S^ 2\) or (ii) \(L_ m\) or \(L_ m'\), the manifolds obtained by Dehn surgeries along an unknotted 2-torus, or (iii) an irrational connected sum along circles of either \(L_ m\) or \(L_ m'\) and \(L(n,r)\times S^ 1\). The author also shows that a 4-manifold obtained by Gluck surgery is also obtained by Dehn surgery along some \(T^ 2\)-knot in \(S^ 4\).
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 4024309
zbMATH Keywords: embedded 2-torus
zbMATH Keywords: embedded 2-torus in the 4-sphere
zbMATH Keywords: spin of a classical torus knot
zbMATH Keywords: Dehn surgery
zbMATH Keywords: 4-manifold obtained by Dehn surgery
zbMATH Keywords: good torus fibration over S 2
zbMATH Keywords: Gluck surgery
MaRDI profile type: Publication
full work available at URL: pjm.1988.133.289
OpenAlex ID: W2083494625
DOI: 10.2140/PJM.1988.133.289
description: scientific article; zbMATH DE number 4024309