{
  "id": "1602.02371",
  "version": "v1",
  "published": "2016-02-07T13:06:41.000Z",
  "updated": "2016-02-07T13:06:41.000Z",
  "title": "Cosmetic surgery and the $SL(2,\\mathbb{C})$ Casson invariant for two-bridge knots",
  "authors": [
    "Kazuhiro Ichihara",
    "Toshio Saito"
  ],
  "comment": "11 pages, 2 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. It is seen that all the two-bridge knots at most 9 crossings other than $9_{27} = S(49,19)=C[2,2,-2,2,2,-2]$ admits no purely cosmetic surgery pairs. Then we show that any two-bridge knot of the Conway form $[2x,2,-2x,2x,2,-2x]$ with $x \\ge 1$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears for $x=1$. Our advantage to prove this is using the $SL(2,\\mathbb{C})$ Casson invariant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-07T13:06:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50",
      "57M25",
      "57M27",
      "57N10"
    ],
    "keywords": [
      "two-bridge knot",
      "casson invariant",
      "cosmetic surgery pairs yielding homology",
      "purely cosmetic surgery pairs",
      "cosmetic surgery problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160202371I"
    }
  }
}