{
  "id": "math/0204057",
  "version": "v1",
  "published": "2002-04-04T05:56:35.000Z",
  "updated": "2002-04-04T05:56:35.000Z",
  "title": "The Lawrence-Krammer representation",
  "authors": [
    "Stephen Bigelow"
  ],
  "comment": "20 pages, 9 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module $H_2(\\tilde{C})$ over the ring of Laurent polynomials in $q$ and $t$. In this paper we describe some surfaces in $\\tilde{C}$ representing elements of homology. We use these to give a new proof that $H_2(\\tilde{C})$ is a free module. We also show that the $(n-2,2)$ representation of the Temperley-Lieb algebra is the image of a map to relative homology at $t=-q^{-1}$, clarifying work of Lawrence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-04-04T05:56:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20C08"
    ],
    "keywords": [
      "lawrence-krammer representation",
      "braid group",
      "temperley-lieb algebra",
      "laurent polynomials",
      "free module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4057B"
    }
  }
}