{
  "id": "1908.08231",
  "version": "v1",
  "published": "2019-08-22T07:16:23.000Z",
  "updated": "2019-08-22T07:16:23.000Z",
  "title": "The Kauffman bracket skein module of the handlebody of genus 2 via braids",
  "authors": [
    "Ioannis Diamantis"
  ],
  "comment": "14 pages, 15 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we present two new bases, $B^{\\prime}_{H_2}$ and $\\mathcal{B}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM($H_2$). We start from the well-known Przytycki-basis of KBSM($H_2$), $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open braid form. We define an ordering relation on an augmented set $L$ consisting of monomials of all different \"loopings\" in $H_2$, that contains the sets $B_{H_2}$, $B^{\\prime}_{H_2}$ and $\\mathcal{B}_{H_2}$ as proper subsets. Using the Kauffman bracket skein relation we relate $B_{H_2}$ to the sets $B^{\\prime}_{H_2}$ and $\\mathcal{B}_{H_2}$ via a lower triangular infinite matrix with invertible elements in the diagonal. The basis $B^{\\prime}_{H_2}$ is an intermediate step in order to reach at elements in $\\mathcal{B}_{H_2}$ that have no crossings on the level of braids, and in that sense, $\\mathcal{B}_{H_2}$ is a more natural basis of KBSM($H_2$). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of c.c.o. 3-manifolds $M$ that are obtained from $H_2$ by surgery, since isotopy moves in $M$ are naturally described by elements in $\\mathcal{B}_{H_2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-22T07:16:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M25",
      "20F36",
      "20F38",
      "20C08"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "handlebody",
      "kauffman bracket skein relation",
      "lower triangular infinite matrix",
      "open braid form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}