A family of two-component links of real projective planes in the four-sphere
type: Item
instance of: scholarly article
title: A family of two-component links of real projective planes in the four-sphere
zbMATH Open document ID: 1028.57022
DOI: 10.1016/S0166-8641(02)00095-0
author: Yuichi Yamada
published in: Topology and its Applications
publication date: 2003-04-28T00:00:00Z
review text: Let \(P_+(P_-)\) denote an equivalence class or its representative of a positive (negative) real projective plane \(P^2\) whose normal Euler number is 2(-2) in \(S^4\) with fixed orientation on \(S^4\). Let \(L = P_+ \cup P_-\) be a two component link of projective planes in \(S^4\) and assume that the Price surgery \(\Pi(P_-)\) along \(P_-\) is diffeomorphic to \(S_4\). Then \(P_+ \subset \Pi(P_-)\) is a positive \(P^2\)-knot in \(S^4\). The obtained pairs \((S^4, P_+)\) and \((\Pi(P_-), P_+)\) may not be diffeomorphic. So, for given positive \(P^2\)-knots \((S^4,P_+)\) and \((S^4, P_+')\), does there exist a negative \(P^2\)-knot \(P_-\) in \(S^4 \setminus P_+\) such that the pairs \((\Pi(P_-),P_+)\) and \((S^4, P_+')\) are diffeomorphic? The author proves that this is true if we restrict ourself to a certain family of \(P^2\)-knots which is described as: \[ [P_{0+} \sharp Tw_2(k)] \mid k \text{ is a classical knot in }S^3, \] where \(P_{0+}\) is the unknotted (standardly embedded) projective plane in \(S^4\), \(Tw_2(k)\) denotes the 2-twist spun 2-knot of a classical knot \(k\) in \(S^3\), the brackets denote the equivalence class and \(\sharp\) the connected sum.
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 1901907
zbMATH Keywords: projective plane knots
zbMATH Keywords: Price surgery
zbMATH Keywords: knotted surface
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00095-0
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description: scientific article; zbMATH DE number 1901907