{
  "id": "1411.5647",
  "version": "v1",
  "published": "2014-11-20T19:29:40.000Z",
  "updated": "2014-11-20T19:29:40.000Z",
  "title": "The SL(2,C) Casson invariant for knots and the $\\widehat{A}$-polynomial",
  "authors": [
    "Hans U. Boden",
    "Cynthia L. Curtis"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper, we extend the definition of the $SL_2(\\Bbb C)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\\widehat{A}$-polynomial of $K$. We prove a product formula for the $\\widehat{A}$-polynomial of the connected sum $K_1 \\# K_2$ of two knots in $S^3$ and deduce additivity of $SL_2(\\Bbb C)$ Casson knot invariant under connected sum for a large class of knots in $S^3$. We also present an example of a nontrivial knot $K$ in $S^3$ with trivial $\\widehat{A}$-polynomial and trivial $SL_2(\\Bbb C)$ Casson knot invariant, showing that neither of these invariants detect the unknot.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-20T19:29:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "casson invariant",
      "polynomial",
      "casson knot invariant",
      "connected sum",
      "product formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}