On $\mu$-Zariski pairs of links

Abstract

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x, y, z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).