{
  "id": "2204.00410",
  "version": "v1",
  "published": "2022-04-01T13:15:38.000Z",
  "updated": "2022-04-01T13:15:38.000Z",
  "title": "The Kauffman bracket skein module of the lens spaces via unoriented braids",
  "authors": [
    "Ioannis Diamantis"
  ],
  "comment": "30 pages, 17 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\\neq 0$. For doing this, we introduce a new concept, that of an {\\it unoriented braid}. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the {\\it generalized Temperley-Lieb algebra of type B}, $TL_{1, n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on $TL_{1, n}$. The universal invariant $V$ is equivalent to the KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description of $L(p,q)$, obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM($L(p,q)$). We first present the solution for the case $q=1$, which corresponds to obtaining a new basis, $\\mathcal{B}_{p}$, for KBSM($L(p,1)$) with $(\\lfloor p/2 \\rfloor +1)$ elements. We note that the basis $\\mathcal{B}_{p}$ is different from the one obtained by Hoste \\& Przytycki. For dealing with the complexity of the infinite system for the case $q>1$, we first show how the new basis $\\mathcal{B}_{p}$ of KBSM($L(p,1)$) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case $q>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-01T13:15:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K31",
      "57K14",
      "20F36",
      "20F38",
      "57K10",
      "57K12",
      "57K45",
      "57K35",
      "57K99",
      "20C08"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "unoriented braid",
      "lens spaces",
      "infinite system",
      "universal kauffman bracket type invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}