---
tag: A82BZ4JQR5
kind: definition
name: Virtual knot
msc: [57K12]
defines: 70ZP456XG6
uses: [QYEAZTYGPT, T3HAGWYKK7]
sources:
  - url: https://en.wikipedia.org/wiki/Virtual_knot
---

## Definition

A *virtual knot diagram* is a generic immersion of an oriented circle in $\R^2$ with marked double points.
Each double point is marked either as a classical crossing, with over- and under-strand, or as a virtual crossing.
Two virtual knot diagrams are *equivalent* if they are related by planar isotopy and a finite sequence of the following moves:

1. the three classical [[Reidemeister moves|T3HAGWYKK7]];
2. the three virtual Reidemeister moves, which are the Reidemeister moves with all crossings virtual;
3. the mixed move, which slides a strand with two virtual crossings past a classical crossing.

A *virtual knot* is an equivalence class of virtual knot diagrams.

## Remarks

The moves of type 3 that slide a strand with classical crossings past a virtual crossing are forbidden.
Example: the virtual trefoil has two classical crossings and one virtual crossing; it is a nontrivial virtual knot that is not classical.
Non-example: the forbidden moves are not equivalences, and allowing both of them makes every virtual knot equivalent to the unknot.
A diagram with no virtual crossings is a diagram of a classical [[knot|QYEAZTYGPT]].
Goussarov, Polyak and Viro showed that classical knots equivalent as virtual knots are isotopic.
Kuperberg showed that virtual knots correspond to knots in thickened oriented surfaces $\Sigma \times [0,1]$, up to stabilization, with a unique irreducible representative.