On surface links whose link groups are abelian
type: Item
instance of: scholarly article
title: On surface links whose link groups are abelian
zbMATH Open document ID: 1298.57018
DOI: 10.1017/S030500411400019X
author: Tetsuya Ito
author: Inasa Nakamura
published in: Mathematical Proceedings of the Cambridge Philosophical Society
publication date: 2014-08-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.2290
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 6333631
zbMATH Keywords: surface link
zbMATH Keywords: link group
zbMATH Keywords: abelian surface link
zbMATH Keywords: torus link
zbMATH Keywords: torus covering torus link
zbMATH Keywords: double linking number
zbMATH Keywords: triple linking number
MaRDI profile type: Publication
arXiv ID: 1302.2290
cites work: Surface links with free abelian groups
cites work: Triple linking numbers and triple point numbers of certain \(T^2\)-links
cites work: Surface links which are coverings over the standard torus
cites work: Q4845496
cites work: A theorem of Sanderson on link bordisms in dimension \(4\)
cites work: Bordism of Links in Codimension 2
cites work: Q3995077
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review text: The surface links considered are images of a smooth embedding of closed oriented surfaces into \(\mathbb R^4\). An abelian surface link \(S\) is a surface link whose link group (i.e. fundamental group of the complement of \(S\)) is abelian. The authors consider 2-links (or \(S^2\)-links) and \(T^2\)-links where each component is a 2-sphere or 2-torus. An \(n\)-component abelian surface link is called a surface link of rank \(n\). The genus \(g(S)\) of \(S\) is the sum of genera of the components of \(S\).NEWLINENEWLINEFirst the authors prove the genus-rank inequality for the considered links. For \(S\) of rank \(n>1\) it holds that: NEWLINE\[NEWLINE n(n-1)< 4g(S), NEWLINE\]NEWLINE which gives a lower bound for the total genus. This gives that the rank of a \(T^2\)-link is at most 4.NEWLINENEWLINEBesides constructing abelian surface links of arbitrary rank, the authors study torus-covering \(T^2\)-links and their link group and prove the double and triple linking number formula for torus-covering \(T^2\)-links. In the final section a variety of examples of abelian surface links is offered.
reviewed by: Ivan Ivanšić
description: scientific article; zbMATH DE number 6333631