{
  "id": "1703.03799",
  "version": "v1",
  "published": "2017-03-10T18:56:09.000Z",
  "updated": "2017-03-10T18:56:09.000Z",
  "title": "DNA Origami and Unknotted A-trails in Torus Graphs",
  "authors": [
    "Ada Morse",
    "William Adkisson",
    "Jessica Greene",
    "David Perry",
    "Brenna Smith",
    "Jo Ellis-Monaghan",
    "Greta Pangborn"
  ],
  "comment": "23 pages, 20 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Motivated by the problem of determining unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterize the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that, aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-10T18:56:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M25",
      "05C45",
      "05C62"
    ],
    "keywords": [
      "dna origami",
      "grids containing unknotted a-trails",
      "checkerboard-colorable torus graph",
      "rectangular grid",
      "infinite families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}