{
  "id": "2307.12275",
  "version": "v1",
  "published": "2023-07-23T09:50:41.000Z",
  "updated": "2023-07-23T09:50:41.000Z",
  "title": "The Kauffman bracket skein module of $S^1\\times S^2$ via braids",
  "authors": [
    "Ioannis Diamantis"
  ],
  "comment": "25 pages, 20 figures. arXiv admin note: substantial text overlap with arXiv:2204.00410",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we present two different ways for computing the Kauffman bracket skein module of $S^1\\times S^2$, ${\\rm KBSM}\\left(S^1\\times S^2\\right)$, via braids. We first extend the universal Kauffman bracket type invariant $V$ for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley-Lieb algebra of type B, to an invariant for knots and links in $S^1\\times S^2$. We do that by imposing on $V$ relations coming from the {\\it braid band moves}. These moves reflect isotopy in $S^1\\times S^2$ and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which, is equivalent to computing ${\\rm KBSM}\\left(S^1\\times S^2\\right)$. We show that ${\\rm KBSM}\\left(S^1\\times S^2\\right)$ is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing ${\\rm KBSM}\\left(S^1\\times S^2\\right)$ via braids. Using this diagrammatic method we also obtain a closed formula for the torsion part of ${\\rm KBSM}\\left(S^1\\times S^2\\right)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-23T09:50:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K31",
      "57K14",
      "20F36",
      "20F38",
      "57K10",
      "57K12",
      "57K45",
      "57K35",
      "57K99",
      "20C08"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "universal kauffman bracket type invariant",
      "diagrammatic method",
      "moves reflect isotopy",
      "solid torus st"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}