Incompressible and pairwise incompressible surfaces in alternating link complements
type: Item
instance of: scholarly article
zbMATH Open document ID: 1246.57021
author: Tomoyo Yoshikawa
author: Kanae Takahashi
publication date: 2012-07-22T00:00:00Z
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 6059354
zbMATH Keywords: alternating link
zbMATH Keywords: incompressible surface
MaRDI profile type: Publication
title: Incompressible and pairwise incompressible surfaces in alternating link complements
published in: Kobe Journal of Mathematics
review text: For a non-split, prime alternating link \(L\) in the \(3\)-sphere, \textit{W. Menasco} [Topology 23, 37--44 (1984; Zbl 0525.57003)] showed that any closed connected incompressible and pairwise incompressible surface \(S\) which intersects \(L\) transversely in a finite number of points has a strong constraint. More precisely, he showed that if the number of points of \(S\cap L\) is less than \(8\), then \(S\) is the \(2\)-sphere, and there are only finitely many isotopy types of \(S\) after fixing the number of points of \(S\cap L\).NEWLINENEWLINEThe main purpose of the paper under review is to examine the case where \(S\cap L\) consists of \(8\) points. The result claims that such a surface \(S\) is either the \(2\)-sphere or a torus. Furthermore, under a mild condition, such surfaces are classified into 16 patterns. In the appendix, two gaps in \textit{W. Menasco} [Pac. J. Math. 117, 353--370 (1985; Zbl 0578.57002)] are filled.
reviewed by: Masakazu Teragaito
description: scientific article; zbMATH DE number 6059354