{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *virtual knot diagram* is a generic immersion of an oriented circle in $\\R^2$ with marked double points.\nEach double point is marked either as a classical crossing, with over- and under-strand, or as a virtual crossing.\nTwo virtual knot diagrams are *equivalent* if they are related by planar isotopy and a finite sequence of the following moves:\n\n1. the three classical [[Reidemeister moves|T3HAGWYKK7]];\n2. the three virtual Reidemeister moves, which are the Reidemeister moves with all crossings virtual;\n3. the mixed move, which slides a strand with two virtual crossings past a classical crossing.\n\nA *virtual knot* is an equivalence class of virtual knot diagrams.\n\n## Remarks\n\nThe moves of type 3 that slide a strand with classical crossings past a virtual crossing are forbidden.\nExample: the virtual trefoil has two classical crossings and one virtual crossing; it is a nontrivial virtual knot that is not classical.\nNon-example: the forbidden moves are not equivalences, and allowing both of them makes every virtual knot equivalent to the unknot.\nA diagram with no virtual crossings is a diagram of a classical [[knot|QYEAZTYGPT]].\nGoussarov, Polyak and Viro showed that classical knots equivalent as virtual knots are isotopic.\nKuperberg showed that virtual knots correspond to knots in thickened oriented surfaces $\\Sigma \\times [0,1]$, up to stabilization, with a unique irreducible representative.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "70ZP456XG6",
    "kind": "definition",
    "msc": [
      "57K12"
    ],
    "name": "Virtual knot",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Virtual_knot"
      }
    ],
    "tag": "A82BZ4JQR5",
    "uses": [
      "QYEAZTYGPT",
      "T3HAGWYKK7"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/A82BZ4JQR5/",
  "links": [
    {
      "field": "defines",
      "target": "70ZP456XG6"
    },
    {
      "field": "uses",
      "target": "QYEAZTYGPT"
    },
    {
      "field": "uses",
      "target": "T3HAGWYKK7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/A82BZ4JQR5.md",
  "msc": [
    "57K12"
  ],
  "name": "Virtual knot",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A82/A82BZ4JQR5.md",
  "status": null,
  "tag": "A82BZ4JQR5",
  "uri": "m2://A82BZ4JQR5",
  "url": "https://math2.tamnd.com/wiki/virtual-knot-definition/",
  "writing": {
    "tag": "A82BZ4JQR5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}