In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about homotopy groups of the complement being isomorphic to $\pi_i (S^1)$ for low values of $i$ (e.g. for $i<n/3$). The cases that I am interested in unfortunately have more complicated fundamental groups. In particular I'm considering ribbon embeddings of $S^n$ and $S^{n-1}\times S^1$ into $S^{n+2}$ with $n>2$. Here 'ribbon' means that the embedding $K$ bounds an immersed solid (a disk for $S^n$ and an immersed $D^n\times S^1$ for $S^{n-1}\times S^1$) which is embedded except for double points, and each component of the double point set lifts to two copies of $D^{n-1}$, one copy lying in the interior and the other having a boundary on the boundary of the solid (this is just the natural generalization to higher dimensions of the definition of 'ribbon' for classical knots, although I've seen the definition given in other equivalent forms). Are there results that could apply to these types of embeddings?
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1$\begingroup$ Perhaps you could clarify what you are actually looking for, eg the last sentence of your question. In other words, does "results" mean the statement that certain invariants could classify such knots (under some hypotheses?) Are you asking for obstructions to a knot being ribbon? The answers to your question yesterday mathoverflow.net/questions/143767/… referred to invariants of knots that are defined for knots without any particular hypotheses, so in particular for ribbon knots. (Some would work for non-spherical knots.) $\endgroup$– Danny RubermanCommented Oct 3, 2013 at 15:58
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$\begingroup$ Some of our techniques arxiv.org/pdf/math/0309150.pdf here might apply in even higher dimensions. $\endgroup$– Scott CarterCommented Oct 3, 2013 at 16:04
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