{
  "id": "2401.01533",
  "version": "v1",
  "published": "2024-01-03T04:03:40.000Z",
  "updated": "2024-01-03T04:03:40.000Z",
  "title": "Twisted Yang-Baxter sets, cohomology theory, and application to knots",
  "authors": [
    "Mohamed Elhamdadi",
    "Manpreet Singh"
  ],
  "comment": "19 pages, 15 figures. Comments are welcome!",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "In this article, we introduce a notion of twisted set-theoretic Yang-Baxter solution, which is a triplet $(X,f,R)$, where $(X,R)$ is a Yang-Baxter set and $f:X \\to X$ is an automorphism of $(X,R)$. We present a cohomology theory for it, and use cocycles of twisted biquandles in amalgamation with Alexander numbering to construct state-sum invariant of knots and knotted surfaces. Additionally, we introduce a twisted version of cohomology theory for Yang-Baxter sets and give applications to knot theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-03T04:03:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K45",
      "57K12",
      "16T25",
      "57T99"
    ],
    "keywords": [
      "cohomology theory",
      "twisted yang-baxter sets",
      "application",
      "twisted set-theoretic yang-baxter solution",
      "construct state-sum invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}