{
  "id": "2306.11001",
  "version": "v1",
  "published": "2023-06-19T15:07:51.000Z",
  "updated": "2023-06-19T15:07:51.000Z",
  "title": "On homology concordance in contractible manifolds and two bridge links",
  "authors": [
    "Hugo Zhou"
  ],
  "comment": "39 pages, 18 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\widehat{\\mathcal{C}}_\\mathbb{Z}$ be the group consists of manifold-knot pairs $(Y,K)$ modulo homology concordance, where $Y$ is an integer homology sphere bounding an integer homology ball, and let $\\mathcal{C}_\\mathbb{Z}$ be the subgroup consisting of pairs $(S^3,K)$. Dai-Hom-Stoffregen-Truong show that the quotient group ${\\widehat{\\mathcal{C}}_\\mathbb{Z}}/{\\mathcal{C}_\\mathbb{Z}}$ admits a $\\mathbb{Z}^\\infty$-summand. In this paper, we improve the result by showing that there exists a family $\\{(Y,K_m)\\}_{m>1 }$ generating the $\\mathbb{Z}^\\infty$-summand where $Y$ is the boundary of a smooth contractible $4$-manifold. In fact, we give a $\\mathbb{Z}$-count of such families. The examples are constructed using a family of knots obtained by blowing down a component of a two-bridge link. They are studied in Jonathan Hales's thesis. Using the algorithm due to Ozsv\\'{a}th, Szab\\'{o} and Hales we give a classification of the knot Floer homology of a larger family of such knots, that might be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-19T15:07:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bridge links",
      "contractible manifolds",
      "integer homology ball",
      "knot floer homology",
      "modulo homology concordance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}