Math 2.0

Manifolds and cell complexes Definition

Virtual knot

Builds on KnotReidemeister's theorem

Definition

A virtual knot diagram is a generic immersion of an oriented circle in ℝ2\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;
  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. 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 Σ×[0,1]\Sigma \times [0,1], up to stabilization, with a unique irreducible representative.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Knot, concept, no evidence levelKnotconceptLink, concept, no evidence levelLinkconceptReidemeister's theorem, statement, no evidence levelReidemeister's theoremstatementVirtual knot, definition, no evidence levelVirtual knotdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Knot concept
  2. Link concept
  3. Reidemeister's theorem statement
  4. Virtual knot definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Virtual_knot
About this page
TagA82BZ4JQR5
URIm2://A82BZ4JQR5
Kinddefinition
WritingW0 stub
MSC57K12
DefinesVirtual knot
UsesKnot, Reidemeister's theorem
Changed2 November 2026

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