On surface links whose link groups are abelian
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.
scientific article; zbMATH DE number 6333631
scientific article