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.