{
  "id": "2405.16607",
  "version": "v1",
  "published": "2024-05-26T15:32:27.000Z",
  "updated": "2024-05-26T15:32:27.000Z",
  "title": "Salter's question on the image of the Burau representation of $B_3$",
  "authors": [
    "Donsung Lee"
  ],
  "comment": "33 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "In 1974, Birman posed the question of under what conditions a matrix with Laurent polynomial entries is in the image of the Burau representation. In 1984, Squier observed that the matrices in the image are contained in a unitary group. In 2021, Salter formulated a specific question: whether the central quotient of the Burau image group is the central quotient of a certain subgroup of the unitary group. We solve this question negatively in the simplest nontrivial case, $n=3$, algorithmically constructing a counterexample.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-26T15:32:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20C99",
      "20E05"
    ],
    "keywords": [
      "burau representation",
      "salters question",
      "central quotient",
      "unitary group",
      "simplest nontrivial case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}