The structures of hypersurface simple K3 singularities

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We show the existence of infinitely many links of non-degenerate simple K3 singularities defined by non-quasi-homogeneous polynomials, which do not appear in the case of the singularities defined by quasi-homogeneous polynomials. Moreover, we give a result that relates the diffeomorphism types of the links of non-degenerate semi-quasi-homogeneous hypersurface simple K3 singularities with the singularities of the normal K3 surface that appears as the exceptional divisor of the resolution of the singularity. We also classify links of the some complex 3-dimensional hypersurface isolated singularities up to regular homotopies under some conditions.