{
  "id": "2405.04337",
  "version": "v1",
  "published": "2024-05-07T14:11:13.000Z",
  "updated": "2024-05-07T14:11:13.000Z",
  "title": "On the Kauffman bracket skein module of $(S^1 \\times S^2) \\ \\# \\ (S^1 \\times S^2)$",
  "authors": [
    "Rhea Palak Bakshi",
    "Seongjeong Kim",
    "Xiao Wang"
  ],
  "comment": "30 pages, 20 figures",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\\mathbb Z[A^{\\pm 1}]$ is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the $3$-manifold $(S^1 \\times S^2) \\ \\# \\ (S^1 \\times S^2)$ over the ring $\\mathbb Z[A^{\\pm 1}]$. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we also compute the Kauffman bracket skein module of $(S^1 \\times S^2) \\ \\# \\ (S^1 \\times D^2)$. Furthermore, we show that the skein module of $(S^1 \\times S^2) \\ \\# \\ (S^1 \\times S^2)$ does not split into the sum of free and $(A^k-A^{-k})$-torsion modules, for each $k\\geq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-07T14:11:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K31",
      "57K10"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "big open problem",
      "non-prime manifolds",
      "laurent polynomials",
      "skein theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}