Some higher dimensional knots (Q1109375)

An n-knot K is of type p if it bounds an \((n+1)\)-disk in a manifold obtained by surgery on a link of \((n-p+1)\)-spheres in the complement of K which is trivial as a link in \(S^{n+2}\). The set \(K_ n(p)\) of ambient isotopy classes of such knots is closed under knot sum. Every ribbon n- knot is of type 1. Using the related notion of disc pair of type p it is shown that the converse also holds. If \(2\leq 2p\leq n\) then a knot of type p is also of type \(n-p+1\). However the converse is false, as it is shown that every 2-knot is of type 2, while there are 2-knots which are not ribbon. These notions are related to properties of Seifert hypersurfaces and ``pseudoribbon maps'' for K. Computations of the Alexander modules of such knots are used to show that \(K_ n(p)\) is not finitely generated.