Surface links with free abelian groups
A surface knot is the image of a smooth embedding of a closed surface in \(\mathbb R^4\). Analogously, a surface link is a smooth embedding of a disjoint union (at least two copies) of a considered surface \(S\). In particular, \(S^2\)-links (resp. \(T^2\)-links) are those whose components are homeomorphic to 2-spheres (resp. tori). The link group is the fundamental group of the link exterior. The author continues his investigation of \(T^2\)-links [\textit{I. Nakamura}, Topology Appl. 159, No. 5, 1439--1447 (2012; Zbl 1236.57030)]. The present paper brings a series of examples of \(T^2\)-links whose link groups are free abelian, something that does not happen with \(S^2\)-links. Previously it was known that a \(T^2\)-link, called ``Hopf 2-link'' has a free abelian group of rank two as its link group. The link groups of the presented \(T^2\)-links are free abelian of rank three. It is also shown that there are infinitely many such \(T^2\)-links. In addition an example is given of \(T^2\)-links whose link group is free abelian of rank four. These \(T^2\)-links are so called ``torus-covering \(T^2\)-links'', defined in the paper.
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