{
  "id": "1702.03479",
  "version": "v1",
  "published": "2017-02-12T02:08:30.000Z",
  "updated": "2017-02-12T02:08:30.000Z",
  "title": "Intrinsic linking with linking numbers of specified divisibility",
  "authors": [
    "Christopher Tuffley"
  ],
  "comment": "16 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $n$, $q$ and $r$ be positive integers, and let $K_N^n$ be the $n$-skeleton of an $(N-1)$-simplex. We show that for $N$ sufficiently large every embedding of $K_N^n$ in $\\mathbb{R}^{2n+1}$ contains a link $L_1\\cup\\cdots\\cup L_r$ consisting of $r$ disjoint $n$-spheres, such that the linking number $link(L_i,L_j)$ is a nonzero multiple of $q$ for all $i\\neq j$. This result is new in the classical case $n=1$ (graphs embedded in $\\mathbb{R}^3$) as well as the higher dimensional cases $n\\geq 2$; and since it implies the existence of a link $L_1\\cup\\cdots\\cup L_r$ such that $|link(L_i,L_j)|\\geq q$ for all $i\\neq j$, it also extends a result of Flapan et al. from $n=1$ to higher dimensions. Additionally, for $r=2$ we obtain an improved upper bound on the number of vertices required to force a two-component link $L_1\\cup L_2$ such that $link(L_1,L_2)$ is a nonzero multiple of $q$. Our new bound has growth $O(nq^2)$, in contrast to the previous bound of growth $O(\\sqrt{n}4^nq^{n+2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-12T02:08:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q45",
      "57M25",
      "57M15"
    ],
    "keywords": [
      "linking number",
      "specified divisibility",
      "intrinsic linking",
      "nonzero multiple",
      "higher dimensional cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}