{
  "id": "2402.04629",
  "version": "v1",
  "published": "2024-02-07T07:24:35.000Z",
  "updated": "2024-02-07T07:24:35.000Z",
  "title": "Iterated satellite operators on the knot concordance group",
  "authors": [
    "Jae Choon Cha",
    "Taehee Kim"
  ],
  "comment": "32 pages, 3 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that for a winding number zero satellite operator $P$ on the knot concordance group, if the axis of $P$ has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an infinite rank subgroup for each $n$. Furthermore, the graded quotients of the filtration of the knot concordance group associated with $P$ have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinz\\'{o}n-Caicedo in many cases. We also show that under the same hypotheses, $P^n$ is not a homomorphism on the knot concordance group for each $n$. We use amenable $L^2$-signatures to prove these results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-07T07:24:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57N70"
    ],
    "keywords": [
      "knot concordance group",
      "iterated satellite operators",
      "winding number zero satellite operator",
      "infinite rank subgroup",
      "blanchfield form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}