{
  "id": "2208.12378",
  "version": "v1",
  "published": "2022-08-25T23:47:55.000Z",
  "updated": "2022-08-25T23:47:55.000Z",
  "title": "On the Burau representation of $B_3$",
  "authors": [
    "Vasudha Bharathram",
    "Joan Birman"
  ],
  "comment": "15 pages, 10 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the Burau representation of the braid group $B_n$ in the case where $n=3$. We give three novel topological proofs that the Burau representation of $B_3$ is faithful, and a proof that it's faithful modulo $p$ for all integers $p>1$. We then classify conjugacy classes in the image of the Burau representation in $\\text{GL}(2, \\mathbb{Z}[t, t^{-1}])$ in a way that takes account of the fact that braids are geometrically oriented, and use that fact to give a new, linear time solution to the conjugacy problem in $B_3$. We pose several open problems about $B_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-25T23:47:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F65",
      "20C99"
    ],
    "keywords": [
      "burau representation",
      "linear time solution",
      "braid group",
      "conjugacy problem",
      "open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}