We introduce the concept of regular genus for n-dimensional links and knots, and study the topological properties of it. 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. Some applications to low-dimensional links and a final section about connected sums of links complete the paper.
A genus for n-dimensional knots and links / Cavicchioli, Alberto. - In: COLLECTANEA MATHEMATICA. - ISSN 0010-0757. - STAMPA. - 36 (3):(1985), pp. 229-242.
A genus for n-dimensional knots and links
CAVICCHIOLI, Alberto
1985
Abstract
We introduce the concept of regular genus for n-dimensional links and knots, and study the topological properties of it. 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. Some applications to low-dimensional links and a final section about connected sums of links complete the paper.
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