A genus for \(n\)-dimensional knots and links (Q1085842)

The concept of regular genus for n-dimensional links is introduced. It extends the classical genus of one-dimensional links. Some characterization theorems of the trivial knot are given. In particular, the only genus zero n-dimensional knot is proved to be homeomorphic with the trivial knot. Then the regular genus of a knot is proved to be related to the one-dimensional homology of the universal abelian covering of its complement. Partial extensions for links of these results are also obtained. Some applications to low-dimensional links and a final section about connected sums of links complete the paper.