7
$\begingroup$

Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the complement is homotopy equivalent to $S^1$. The question which I wonder is, does it suffice to show that the fundamental group is $\mathbb{Z}$, given the fact that $K$ is ribbon? For $n=2$ this would be true, as Freedman mentions (that having infinite cyclic fundamental group implies being homotopy equivalent to a circle) but I am not terribly familiar with the high-dimensional case.

$\endgroup$
7
$\begingroup$

If the fundamental group of the complement is the integers there's still the various Alexander modules for the complement. Ribbon knots aren't immune from having non-trivial Alexander modules in dimensions higher than 1. Take a look at Kearton's papers on "simple n-knots"

$\endgroup$
2
$\begingroup$

I found a paper which discusses this issue: Ribbon knots and ribbon disks from Asano, Marumoto, and Yanagawa. They establish that for $n\geq 3$ a ribbon knot with infinite cyclic fundamental group is trivial.

http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm18_01_12.pdf

$\endgroup$
  • $\begingroup$ On the other hand Suciu claims that their result is based on an unproved result erroneously attributed to Lomonaco. maths.ed.ac.uk/~aar/papers/suciu1.pdf $\endgroup$ – Blake Jun 28 '12 at 12:33
  • $\begingroup$ I guess I did make that claim, back in the day. Surely I must have had a reason, but I forgot by now what that reason was. Nevertheless, that paper of S.J. Lomonaco, cited by Asano, Marumoto, and Yanagawa as "The homotopy groups of knots, II. A solution to Problem 36 of R.H. Fox, to appear", is not listed on MathSciNet, and I could not find a reference to it on GoogleScholar, either. $\endgroup$ – Alex Suciu Jul 4 '12 at 23:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.