A family of two-component links of real projective planes in the four-sphere
Let \(P_+(P_-)\) denote an equivalence class or its representative of a positive (negative) real projective plane \(P^2\) whose normal Euler number is 2(-2) in \(S^4\) with fixed orientation on \(S^4\). Let \(L = P_+ \cup P_-\) be a two component link of projective planes in \(S^4\) and assume that the Price surgery \(\Pi(P_-)\) along \(P_-\) is diffeomorphic to \(S_4\). Then \(P_+ \subset \Pi(P_-)\) is a positive \(P^2\)-knot in \(S^4\). The obtained pairs \((S^4, P_+)\) and \((\Pi(P_-), P_+)\) may not be diffeomorphic. So, for given positive \(P^2\)-knots \((S^4,P_+)\) and \((S^4, P_+')\), does there exist a negative \(P^2\)-knot \(P_-\) in \(S^4 \setminus P_+\) such that the pairs \((\Pi(P_-),P_+)\) and \((S^4, P_+')\) are diffeomorphic? The author proves that this is true if we restrict ourself to a certain family of \(P^2\)-knots which is described as: \[ [P_{0+} \sharp Tw_2(k)] \mid k \text{ is a classical knot in }S^3, \] where \(P_{0+}\) is the unknotted (standardly embedded) projective plane in \(S^4\), \(Tw_2(k)\) denotes the 2-twist spun 2-knot of a classical knot \(k\) in \(S^3\), the brackets denote the equivalence class and \(\sharp\) the connected sum.
scientific article; zbMATH DE number 1901907
scientific article