{
  "id": "1710.05237",
  "version": "v1",
  "published": "2017-10-14T20:54:17.000Z",
  "updated": "2017-10-14T20:54:17.000Z",
  "title": "Unknotting numbers for prime $θ$-curves up to seven crossings",
  "authors": [
    "Dorothy Buck",
    "Danielle O'Donnol"
  ],
  "comment": "14 pages, 7 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Determining unknotting numbers is a large and widely studied problem. We consider the more general question of the unknotting number of a spatial graph. We show the unknotting number of spatial graphs is subadditive. Let $g$ be an embedding of a planar graph $G$, then we show $u(g) \\geq \\max\\{u(s) | s$ is a non-overlapping set of constituents of $g\\}$. Focusing on $\\theta$-curves, we determine the exact unknotting numbers of the $\\theta$-curves in the Litherland-Moriuchi Table, excepting $\\mathbf{7_5,7_{22}, 7_{24}},$ and $\\mathbf{7_{58}}$ which have unknotting number $1$ or $2$. Additionally, we demonstrate unknotting crossing changes for all of the curves. In doing this we introduce new methods for obstructing unknotting number $1$ in $\\theta$-curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-14T20:54:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M15"
    ],
    "keywords": [
      "seven crossings",
      "spatial graph",
      "general question",
      "planar graph",
      "exact unknotting numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}