{
  "id": "2407.01729",
  "version": "v1",
  "published": "2024-07-01T19:03:00.000Z",
  "updated": "2024-07-01T19:03:00.000Z",
  "title": "Deformations of curves with constant curvature",
  "authors": [
    "Mohammad Ghomi",
    "Matteo Raffaelli"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "math.DG",
    "math.GT"
  ],
  "abstract": "We prove that curves of constant curvature satisfy the parametric $C^1$-dense relative $h$-principle in the space of immersed curves with nonvanishing curvature in Euclidean space $R^{n\\geq 3}$. It follows that two knots of constant curvature in $R^3$ are isotopic, resp. homotopic, through curves of constant curvature if and only if they are isotopic, resp. homotopic, and their self-linking numbers, resp. self-linking numbers mod $2$, are equal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-01T19:03:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A04",
      "57K10",
      "58C35",
      "53C21"
    ],
    "keywords": [
      "deformations",
      "constant curvature satisfy",
      "self-linking numbers mod",
      "euclidean space",
      "immersed curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}