---
tag: 70ZP456XG6
kind: concept
name: Virtual knot
msc: [57K12]
sources:
  - url: https://en.wikipedia.org/wiki/Virtual_knot
same_as:
  wikidata: Q7935129
  wikipedia: Virtual knot
---

## Summary

A virtual knot is an equivalence class of knot diagrams that may have, besides ordinary crossings, virtual crossings that are not really there.
Louis Kauffman introduced them as a generalization of classical [[knots|QYEAZTYGPT]].
Geometrically, virtual knots are knots in thickened surfaces, taken up to a suitable stabilization.
Classical knots form a part of virtual knot theory: two classical knots that are equivalent as virtual knots are equivalent as classical knots.

## Explanations

### L1 human

Some knot drawings cannot be made without extra crossings that come only from the flat paper.
A virtual crossing is such a fake crossing, drawn with a small circle around it.
Virtual knots allow these crossings and treat them as invisible: strands may slide freely across them.
They can be pictured as knots drawn on a surface with handles, such as a torus, where the fake crossings disappear.