{
  "id": "enwiki-001322644662",
  "slug": "boundary-parallel",
  "title": "Boundary parallel",
  "qid": "",
  "lead": "In mathematics, a connected submanifold of a compact manifold with boundary is said to be boundary parallel, ∂-parallel, or peripheral if it can be continuously deformed into a boundary component. This notion is important for 3-manifold topology.",
  "body": "In mathematics, a connected submanifold of a compact manifold with boundary is said to be boundary parallel, ∂-parallel, or peripheral if it can be continuously deformed into a boundary component. This notion is important for 3-manifold topology. ==Boundary-parallel embedded surfaces in 3-manifolds== If F is an orientable closed surface smoothly embedded in the interior of an manifold with boundary M then it is said to be boundary parallel if a connected component of M \\smallsetminus F is homeomorphic to F \\smallsetminus [0, 1[ . In general, if (F, \\partial F) is a topologically embedded compact surface in a compact 3-manifold (M, \\partial M) some more care is needed: one needs to assume that F admits a bicollar, and then F is boundary parallel if there exists a subset P \\subset M such that F is the frontier of P in M and P is homeomorphic to F \\times [0, 1] . ==Context and applications== ==See also== Atoroidal Satellite knot ==References==",
  "categories": [
    "102487739",
    "104909798",
    "107702204",
    "20156305",
    "23118434",
    "23603337",
    "23888906",
    "24043274"
  ],
  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1322644662",
  "content_hash": "",
  "created_at": null,
  "updated_at": 1763370222.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
  "csl": {
    "id": "enwiki-001322644662",
    "type": "webpage",
    "title": "Boundary parallel",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/boundary-parallel",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "accessed": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001322644662",
    "isBasedOn": "https://en.wikipedia.org/w/index.php?oldid=1322644662",
    "note": "Immutable snapshot enwiki-001322644662"
  }
}