Spinning, twisting, and branched covers

Twist-roll spun knots are a well-studied family of knotted 2-spheres in the 4-sphere. The inputs for determining a twist roll spun knots are an integer number of twists, a number of rolls, and a choice of classical knot in the 3-sphere. I will explain why adding twists, while changing the resulting 2-sphere, preserves certain cyclic branched covers of the 4-sphere. As a consequence, we’ll see that the double branched covers branched along the exotic projective planes recently constructed by Miyazawa are all diffeomorphic to standard CP^2. This is joint with Mark Hughes and Seungwon Kim.