13
$\begingroup$

In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare conjecture.

In a series of papers, Akbulut and Gompf have shown most of these Cappell-Shaneson knots actually are knots in the standard $S^4$, the most recent reference being this.

In principle, one should be able to work through their arguments to derive a picture of these 2-knots in the 4-sphere. Has anyone done this, for any of the Cappell-Shaneson knots?

I know various people have created censi of 2-knots, does anyone know if any Cappell-Shaneson knots appear in those censi? (I have a hard time accepting censuses as plural of census, sorry, it sounds so wrong!)

I'd be happy with any fairly explicit geometric picture of a Cappell-Shaneson knot sitting in $S^4$. The two I'm most familiar with is the Whitneyesque motion-diagram, and the "resolution of a knotted 4-valent graph in $S^3$" picture. What I want to avoid is the "attach a handle and fuss about and argue that the manifold you've constructed is diffeomorphic to $S^4$" situation.

$\endgroup$

2 Answers 2

8
$\begingroup$

There is a paper by Iain Aitcheson (possible mis-spelling of the last name) and Hyam Rubenstein published in a Contemporary Mathematics Series of the AMS (Conference Proceedings) that is the most explicit description of which I know. I wanted to to try and draw the corresponding knot diagrams or Yoshikawa diagrams at one time, but never found the time or engery for it. It is a pity.

Daniel Nash may have a paper about this on the ArXiv. Yep, here and here . I am sorry but I don't have mathscinet at home to look up the reference for the first example.

$\endgroup$
2
  • $\begingroup$ Thanks Scott. I got onto MathSciNet today and couldn't find the Rubinstein-Aitchison paper. They have quite a few papers together so it wasn't clear which one it might be in. $\endgroup$ Jul 24, 2011 at 5:20
  • $\begingroup$ Ryan: It is this one, MR0780575 (86h:57014) Aitchison, I. R.; Rubinstein, J. H. Fibered knots and involutions on homotopy spheres. Four-manifold theory (Durham, N.H., 1982), 1–74, Contemp. Math., 35, Amer. Math. Soc., Providence, RI, 1984. (Reviewer: Selman Akbulut) $\endgroup$ Jul 25, 2011 at 21:47
5
$\begingroup$

I think the explicit embedding of Cappell-Shaneson knot is given in the following paper:

S. Akbulut and R. Kirby, A potential smooth counterexample to in dimension 4 to the Poincare conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology 24 (1985) 375--390. (See Figure 16 of that paper)

The paper of Aitchison and Rubinstein mentioned by Scott Carter figures out that there is an error (on the $\mathbb{Z}/2$-framing of $\gamma$-curve which turns out to be 1) in S. Akbulut and R. Kirby's former paper "An exotic involution on $S^4$, Topology 18 (1979) 1--15. Hence, what S. Akbulut and Kirby really showed (in 1979) is that the specific (or the simplest) Cappell-Shaneson sphere is obtained from the Gluck construction of a smooth 2-knot in standard $S^4$. Figure 16 of 1985 topology paper of S. Akbulut and R. Kirby describes that a smooth 2-knot is obtained from gluing two ribbon disks of a knot $8_9$.

Finally, I would like to say that there is a same stuff given in Figure 6.2, page 17 of Kirby's famous book "The topology of 4-manifolds" Springer Lecture notes in Mathematics 1374.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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