---
tag: QYEAZTYGPT
kind: concept
name: Knot
msc: [57K10]
sources:
  - url: https://en.wikipedia.org/wiki/Knot_(mathematics)
same_as:
  wikidata: Q1188853
  wikipedia: Knot (mathematics)
---

## Summary

A knot is a closed curve in three-dimensional space that does not cross itself, considered up to continuous deformation of the surrounding space.
Formally it is an embedding of the [[circle|060W5CBZP6]] in $\R^3$ or in the $3$-sphere.
Knot theory asks when two knots are equivalent, and answers it with [[knot invariants|GZQ5ZE3X6G]] such as the [[knot group|AQVHM3FP0D]] and knot polynomials.
Knots are also the basic examples in the study of [[3-manifolds|R20TB2KEQQ]].

## Explanations

### L1 human

Take a piece of string, tie it in some way and then glue its two ends together.
The result is a knot: one can move it around in space, but one cannot pass the string through itself or cut it.
A round circle is the [[unknot|1TQ06R4Q66]], the simplest knot.
The [[trefoil|VYTYSVHN4A]], with three crossings, is the simplest knot that cannot be undone.