{
  "id": "1405.1947",
  "version": "v2",
  "published": "2014-05-08T14:43:07.000Z",
  "updated": "2015-12-05T03:18:30.000Z",
  "title": "Lin-Wang type formula for the Haefliger invariant",
  "authors": [
    "Keiichi Sakai"
  ],
  "comment": "25 pages, 10 figures (published version)",
  "journal": "Homology, Homotopy and Applications 17 (2015), no. 2, 317-341",
  "doi": "10.4310/HHA.2015.v17.n2.a15",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "In this paper we study the Haefliger invariant for long embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\\mathbb{R}^{4k-1}\\looparrowright\\mathbb{R}^{6k-1}$. We define the notion of \"crossing changes\" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\\mathbb{R}^{4k-1}$. This formula is a higher-dimensional analogue to that of X.-S. Lin and Z. Wang for the order $2$ invariant for classical knots. As a consequence, we show that the Haefliger invariant is of order two in a similar sense to Birman and Lin. We also give an alternative proof for the result of M. Murai and K. Ohba concerning \"unknotting numbers\" of embeddings $\\mathbb{R}^3\\hookrightarrow\\mathbb{R}^6$. Our formula enables us to define an invariant for generic long immersions $\\mathbb{R}^{4k-1}\\looparrowright\\mathbb{R}^{6k-1}$ which are liftable to embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$. This invariant corresponds to V. Arnold's plane curve invariant in Lin-Wang theory, but in general our invariant does not coincide with order $1$ invariant of T. Ekholm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-08T14:43:07.000Z",
      "title": "Lin-Wang type formula for Haefliger invariant",
      "abstract": "In this paper we study Haefliger invariant for long embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$ in terms of the self-intersections of their generic projections to $\\mathbb{R}^{6k-1}$. We define the notion of \"crossing changes\" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\\mathbb{R}^{4k-1}$. This formula is a higher dimensional analogue to that of X.-S. Lin and Z. Wang for the order two invariant for classical knots, and as a consequence we show that Haefliger invariant behaves like it is of order two. We also give an alternative proof for a result of M. Murai and K. Ohba concerning \"unknotting numbers\" of embeddings $\\mathbb{R}^3\\hookrightarrow\\mathbb{R}^6$. Our formula enables us to define an invariant for generic long immersions $\\mathbb{R}^{4k-1}\\looparrowright\\mathbb{R}^{6k-1}$ which are liftable to embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$. This invariant corresponds to V. Arnold's plane curve invariant in Lin-Wang theory, but in general our invariant does not coincide with order one invariant of T. Ekholm.",
      "comment": "18 pages, 9 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-05T03:18:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58D10",
      "81Q30",
      "57R40",
      "57R42"
    ],
    "keywords": [
      "lin-wang type formula",
      "embeddings",
      "arnolds plane curve invariant",
      "crossing changes",
      "generic long immersions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1947S"
    }
  }
}