{
  "id": "2010.14030",
  "version": "v1",
  "published": "2020-10-27T03:23:20.000Z",
  "updated": "2020-10-27T03:23:20.000Z",
  "title": "A characterization of alternating links in thickened surfaces",
  "authors": [
    "Hans U. Boden",
    "Homayun Karimi"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We use an extension of Gordon-Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\\Sigma \\times I$, then the Gordon-Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$. A compact surface in $\\Sigma \\times I$ is called \\emph{definite} if its Gordon-Litherland pairing is a definite form. Our main result is that a non-split link $L$ in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-27T03:23:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "thickened surface",
      "alternating links",
      "definite form",
      "symmetric bilinear",
      "first homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}