{
  "id": "1611.08001",
  "version": "v1",
  "published": "2016-11-23T21:17:56.000Z",
  "updated": "2016-11-23T21:17:56.000Z",
  "title": "On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology",
  "authors": [
    "Andrew Manion"
  ],
  "comment": "60 pages; 11 figures",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "We relate decategorifications of Ozsv\\'ath-Szab\\'o's new bordered theory for knot Floer homology to representations of $\\mathcal{U}_q(\\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\\mathcal{C}_r(n,\\mathcal{S})$ and $\\mathcal{C}_l(n,\\mathcal{S})$ of Ozsv\\'ath- Szab\\'o's algebra $\\mathcal{B}(n,\\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $\\mathcal{U}_q(\\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsv\\'ath-Szab\\'o's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsv\\'ath-Szab\\'o's theory and Viro's quantum relative $\\mathcal{A}^1$ of the Reshetikhin-Turaev functor based on $\\mathcal{U}_q(\\mathfrak{gl}(1|1))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-23T21:17:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "17B37"
    ],
    "keywords": [
      "knot floer homology",
      "szabós bordered theory",
      "ozsvath-szabos da bimodules",
      "grothendieck groups",
      "tensor products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}