{
  "id": "2307.08605",
  "version": "v1",
  "published": "2023-07-17T16:14:52.000Z",
  "updated": "2023-07-17T16:14:52.000Z",
  "title": "Knot groups, quandle extensions and orderability",
  "authors": [
    "Idrissa Ba",
    "Mohamed Elhamdadi"
  ],
  "comment": "16 pages, comments are welcome!",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "This paper gives a new way of characterizing L-space $3$-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the $n$-quandle $Q_n(L)$ of the link quandle of $L$ is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the $p$-enveloping group of the link quandle is right circularly orderable for any prime integer $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-17T16:14:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M07",
      "57M27",
      "06F15",
      "20F99"
    ],
    "keywords": [
      "knot groups",
      "quandle extensions",
      "link quandle",
      "right circular orderability",
      "right circularly orderable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}