{
  "id": "2202.00238",
  "version": "v1",
  "published": "2022-02-01T06:33:52.000Z",
  "updated": "2022-02-01T06:33:52.000Z",
  "title": "$\\mathfrak{gl}(1 \\vert 1)$-Alexander polynomial for $3$-manifolds",
  "authors": [
    "Yuanyuan Bao",
    "Noboru Ito"
  ],
  "comment": "18 pages. Comments are welcome",
  "categories": [
    "math.GT"
  ],
  "abstract": "As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\\mathfrak{gl}(1\\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-01T06:33:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K16",
      "57K31"
    ],
    "keywords": [
      "alexander polynomial",
      "non-semisimple ribbon tensor categories",
      "manifold invariant",
      "distinguish homotopy equivalent manifolds",
      "turaevs invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}