{
  "id": "1904.05288",
  "version": "v1",
  "published": "2019-04-10T16:44:00.000Z",
  "updated": "2019-04-10T16:44:00.000Z",
  "title": "Concordances to prime hyperbolic virtual knots",
  "authors": [
    "Micah Chrisman"
  ],
  "comment": "32 pages, 25 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\Sigma_0,\\Sigma_1$ be closed oriented surfaces. Two oriented knots $K_0 \\subset \\Sigma_0 \\times [0,1]$ and $K_1 \\subset \\Sigma_1 \\times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \\times [0,1]$ such that $\\partial W=\\Sigma_1 \\sqcup -\\Sigma_0$ and $\\partial A=K_1 \\sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \\subset \\Sigma \\times [0,1]$ admits a nontrivial decomposition along a separating vertical annulus that intersects $K$ in two points. Here we prove that every knot $K \\subset \\Sigma \\times [0,1]$ is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in $\\Sigma \\times [0,1]$, we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby-Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in $\\Sigma \\times [0,1]$ is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in $3$-balls we call complementary tangles. Properties of complementary tangles are studied in detail.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-10T16:44:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "prime hyperbolic virtual knots",
      "concordance",
      "complementary tangles",
      "virtual knots lies",
      "prime satellite knot"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}