Proposed Property 2R counterexamples examined

Abstract

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 analysi 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.