{
  "id": "2203.10684",
  "version": "v1",
  "published": "2022-03-21T00:34:05.000Z",
  "updated": "2022-03-21T00:34:05.000Z",
  "title": "On $ μ$-Zariski pairs of links",
  "authors": [
    "Mutsuo Oka"
  ],
  "comment": "4 figures",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "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}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-21T00:34:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S55"
    ],
    "keywords": [
      "zeta function",
      "projective curves",
      "weak zariski pairs",
      "first homology",
      "isolated hypersurface singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}