More Exotic ${\mathbb{R}\mathbb{P}}^2$-knots and Homotopy Spheres

We extend the infinite family of exotic embeddings ${\mathbb{R}\mathbb{P}}^2\hookrightarrow S^4$ constructed by Miyazawa to a strictly larger family of exotic embeddings, by showing that in place of the pretzel knot $P(-2,3,7)$, an infinite family of knots may be used as input to the construction. To this end, we prove that for any Montesinos knot of the form $K(2,3,\mathrm{\mid }6s+1\mathrm{\mid })$, the branched double cover of the corresponding roll-spun knot is a homotopy sphere. This in turn produces a larger family of homotopy spheres and homotopy ${\mathbb{C}\mathbb{P}}^2$s with potentially interesting involutions. We also observe that Miyazawa’s homotopy sphere can be obtained from $S^4$ by a Gluck twist.