For every integer s, the branched double cover of the roll-spun Montesinos knot K(2,3,|6s+1|) is a homotopy 4-sphere, giving infinitely many exotic topologically unknotted RP^2-knots in S^4 with all odd Seiberg-Witten degrees.
Gluck twist and unknotting of satellite $2$-knots
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abstract
In this paper, we show that the Gluck twist of certain satellite $2$-knots in a $4$-manifold do not change the diffeomorphism type in three different ways: one is directly from the definition of the satellite $2$-knot, and the other two are by finding an equivalent description of the satellite $2$-knot. Furthermore, using the new description, we gave infinite number of new examples of $2$-knots which are unknotted by connected summing a single standard real projective plane.
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More Exotic $\mathbb{RP}^2$-knots and Homotopy Spheres
For every integer s, the branched double cover of the roll-spun Montesinos knot K(2,3,|6s+1|) is a homotopy 4-sphere, giving infinitely many exotic topologically unknotted RP^2-knots in S^4 with all odd Seiberg-Witten degrees.