{
  "id": "2012.09584",
  "version": "v1",
  "published": "2020-12-17T13:53:30.000Z",
  "updated": "2020-12-17T13:53:30.000Z",
  "title": "Shifting chain maps in quandle homology and cocycle invariants",
  "authors": [
    "Yu Hashimoto",
    "Kokoro Tanaka"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\\sigma$ on each quandle chain complex that lowers the dimensions by one. By using its pull-back $\\sigma^\\#$, each $2$-cocycle $\\phi$ gives us the $3$-cocycle $\\sigma^\\# \\phi$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $\\phi$ and their shadow $3$-cocycle invariants associated with $\\sigma^\\# \\phi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $\\sigma^\\# \\phi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-17T13:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K12",
      "57K10",
      "57K45",
      "55N99"
    ],
    "keywords": [
      "cocycle invariants",
      "shifting chain map",
      "quandle chain complex",
      "quandle homology theory",
      "homology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}