{
  "id": "1710.01745",
  "version": "v1",
  "published": "2017-10-04T18:01:31.000Z",
  "updated": "2017-10-04T18:01:31.000Z",
  "title": "Bridge trisections of knotted surfaces in 4--manifolds",
  "authors": [
    "Jeffrey Meier",
    "Alexander Zupan"
  ],
  "comment": "17 pages, 5 figures. Comments welcome",
  "categories": [
    "math.GT"
  ],
  "abstract": "We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \\emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \\emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside $\\mathbb{CP}^2$. Using these examples, we prove that there exist exotic 4--manifolds with $(g,0)$--trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-04T18:01:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bridge trisection",
      "knotted surfaces",
      "complete diagrammatic calculus",
      "conjectural uniqueness result",
      "surface meets components"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}