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

TL;DR

This paper expands the family of exotic embeddings of real projective planes in four-dimensional spheres and constructs new homotopy spheres using Montesinos knots, revealing novel involutions and relationships with Gluck twists.

Contribution

It generalizes Miyazawa's construction to a broader class of knots, producing more exotic embeddings and homotopy spheres with potential symmetries.

Findings

Infinite family of exotic $ ext{RP}^2$ embeddings in $S^4$ constructed.

Branched double covers of certain roll-spun knots are homotopy spheres.

Miyazawa's homotopy sphere can be obtained via a Gluck twist.

Abstract

We extend the infinite family of exotic embeddings $\mathbb{RP}^2 \hookrightarrow S^4$ 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 $K(2,3,|6s+1|)$, 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 $\mathbb{CP}^2$s with potentially interesting involutions. We also observe that Miyazawa's homotopy sphere can be obtained from $S^4$ by a Gluck twist.