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.
