{
  "id": "2406.17454",
  "version": "v1",
  "published": "2024-06-25T10:49:22.000Z",
  "updated": "2024-06-25T10:49:22.000Z",
  "title": "On torsion in the Kauffman bracket skein module of $3$-manifolds",
  "authors": [
    "Giulio Belletti",
    "Renaud Detcherry"
  ],
  "comment": "37 pages, 16 figures. Comments welcome!",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\\mathcal{S}(M)$ has torsion as a $\\mathbb{Z}[A^{\\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\\mathcal{S}(\\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-25T10:49:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K31"
    ],
    "keywords": [
      "kauffman bracket skein module",
      "sided non-boundary parallel essential surface",
      "study kirby problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}