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](https://arxiv.org/abs/0908.1914).


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.



Explicit embeddings of Cappell-Shaneson knots
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](http://arxiv.org/pdf/1103.5571)  and  [here](http://arxiv.org/pdf/1101.2981) . I am sorry but I don't have mathscinet at home to look up the reference for the first example. 


