Links of singularities up to regular homotopy

Abstract: The abstract link L_d of the complex isolated singularity x^2 + y^2 + z^2 + v^{2d} = 0 is diffeomorphic to S^3 \times S^2. We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S^3 \times S^2. Let us denote by i_d the inclusion of L_d in S^7. We show that for arbitrary diffeomorphisms \varphi_d of S^3 \times S^2 with L_d the compositions i_d \circ \varphi_d are image regularly homotopic for two values d_1 and d_2 if and only if d_1-d_2 is even.