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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.

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2 Answers 2

up vote 7 down vote accepted

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.

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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. –  Ryan Budney Jul 24 '11 at 5:20
    
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) –  Scott Carter Jul 25 '11 at 21:47

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.

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