{
  "id": "1501.02504",
  "version": "v1",
  "published": "2015-01-11T22:36:16.000Z",
  "updated": "2015-01-11T22:36:16.000Z",
  "title": "A class of knots with simple $SU(2)$ representations",
  "authors": [
    "Raphael Zentner"
  ],
  "comment": "22 pages, 10 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of a 2-bridge knot. Pretzel knots with bridge number $\\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\\det(K)$. In fact, the complex underlying $I^\\natural(K)$ is of rank equal to $\\det(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat to our surprise, these knots are alternating. Furthermore, with the methods we use, we show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-11T22:36:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "representations",
      "strong l-spaces",
      "fundamental group",
      "rank equal",
      "simple knot"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150102504Z"
    }
  }
}