A family of two-component links of real projective planes in the four-sphere

We are concerned with two-component links of real projective planes in the four sphere, and a surgery called the “Price surgery” along one component changing the other component. Using a certain circle-action on the 4-sphere, we construct a family of knotted projective planes and show that for any given pair P and P′ in the family, P is changed into P′ by a Price surgery along a projective plane in the exterior of P. We also give alternative simple proofs to some known facts and show related lemmas on two-component links of projective planes.