{
  "id": "2106.04965",
  "version": "v1",
  "published": "2021-06-09T10:21:02.000Z",
  "updated": "2021-06-09T10:21:02.000Z",
  "title": "The Kauffman bracket skein module of the complement of $(2, 2p+1)$-torus knots via braids",
  "authors": [
    "Ioannis Diamantis"
  ],
  "comment": "27 pages, 18 figures",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "In this paper we compute the Kauffman bracket skein module of the complement of $(2, 2p+1)$-torus knots, $KBSM(T_{(2, 2p+1)}^c)$, via braids. We start by considering geometric mixed braids in $S^3$, the closure of which are mixed links in $S^3$ that represent links in the complement of $(2, 2p+1)$-torus knots, $T_{(2, 2p+1)}^c$. Using the technique of parting and combing, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group $B_{2, n}$ and that are followed by their ``coset'' part, that represents $T_{(2, 2p+1)}^c$. In that way we show that links in $T_{(2, 2p+1)}^c$ may be pushed to the genus 2 handlebody, $H_2$, and we establish a relation between $KBSM(T_{(2, 2p+1)}^c)$ and $KBSM(H_2)$. In particular, we show that in order to compute $KBSM(T_{(2, 2p+1)}^c)$ it suffices to consider a basis of $KBSM(H_2)$ and study the effect of combing on elements in this basis. We consider the standard basis of $KBSM(H_2)$ and we show how to treat its elements in $KBSM(T_{(2, 2p+1)}^c)$, passing through many different spanning sets for $KBSM(T_{(2, 2p+1)}^c)$. These spanning sets form the intermediate steps in order to reach at the set $\\mathcal{B}_{T_{(2, 2p+1)}^c}$, which, using an ordering relation and the notion of total winding, we prove that it forms a basis for $KBSM(T_{(2, 2p+1)}^c)$. We finally consider c.c.o. 3-manifolds $M$ obtained from $S^3$ by surgery along the trefoil knot and we discuss steps needed in order to compute the Kauffman bracket skein module of $M$. We first demonstrate the process described before for computing the Kauffman bracket skein module of the complement of the trefoil, $KBSM(Tr^c)$, and we study the effect of braid band moves on elements in the basis of $KBSM(Tr^c)$. These moves reflect isotopy in $M$ and are similar to the second Kirby moves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-09T10:21:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K12",
      "57K14",
      "57K35",
      "57K45",
      "57K99",
      "20F36",
      "20F38",
      "20C08"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "torus knots",
      "complement",
      "mixed braid",
      "spanning sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}