Branched covers of twist-roll spun knots and turned twisted tori

Abstract

We prove that the double branched cover of a twist-roll spun knot in $S^4$ is smoothly preserved when four twists are added, and that the double branched cover of a twist-roll spun knot connected sum with a trivial projective plane is preserved after two twists are added. As a consequence, we conclude that the members of a family of homotopy ${\mathbb{CP}}^2$s recently constructed by Miyazawa are each diffeomorphic to ${\mathbb{CP}}^2$. We also apply our techniques to show that the double branched covers of odd-twisted turned tori are all diffeomorphic to $S^2\times S^2$, and show that a family of homotopy 4-spheres constructed by Juhász and Powell are all diffeomorphic to $S^4$.

New results have been added regarding the double branched covers of turned twisted tori, mainly centered around Theorems 1.9 to 1.11. This version is 15 pages with 7 figures. Comments welcome!