On $ \mu$-Zariski pairs of links

TL;DR

This paper introduces a new concept of Zariski pairs for links of isolated hypersurface singularities, providing examples with identical invariants but different topological types, and explores their diffeomorphism properties.

Contribution

It extends Zariski pair theory to links of hypersurface singularities, constructing examples with identical invariants yet topologically distinct links.

Findings

New examples of Zariski pairs with same μ* sequence and zeta function

Links of hypersurface pairs can be non-diffeomorphic despite identical invariants

Hypersurface pairs from Zariski pairs can produce diffeomorphic links

Abstract

The notion of Zariski pairs for projective curves in $\mathbb P^2$ is known since the pioneer paper of Zariski \cite{Zariski} and for further development, we refer the reference in \cite{Bartolo}.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 same Milnor number (\cite{Almost}). We give new examples of Zariski pairs which have the same $\mu^*$ sequence and a same zeta function but two functions belong to different connected components of $\mu$-constant strata (Theorem \ref{mu-zariski}). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology which implies the Jordan form of their monodromies are different (Theorem \ref{main2}). 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 hypersurface pair constructed from a Zariski pair give a diffeomorphic links (Theorem \ref{main3}).