In 1985, Akbulut and Kirby analyzed a homotopy 4-sphere __\( \Sigma \)__ that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysis showed that __\( \Sigma \)__ was one of an infinite collection of examples, all of which were (sadly) the standard __\( S^4 \)__, but with an unusual handle structure.

Recent work with Gompf and Thompson, showed that the construction gives rise to a family __\( L_n \)__ of 2-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each __\( L_n \)__, one component is the simple square knot __\( Q \)__, and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the __\( L_n \)__. In view of the continuing interest and potential importance of the Cappell–Shaneson–Akbulut–Kirby–Gompf examples (e.g., the original __\( \Sigma \)__ is known to embed very efficiently in __\( S^4 \)__ and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.