---
tag: 1TQ06R4Q66
kind: example
name: Unknot
msc: [57K10]
instance_of: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Unknot
same_as:
  wikidata: Q1188344
  wikipedia: Unknot
---

## Example

The *unknot* is the round circle
$$U = \set{(x, y, 0) \in \R^3 : x^2 + y^2 = 1},$$
the image of the map $t \mapsto (\cos t, \sin t, 0)$, which is continuous and injective on the [[circle|060W5CBZP6]] $\R / 2\pi\Z$.
So $U$ is a [[knot|QYEAZTYGPT]], and it is tame since it is equivalent to a square, a closed polygon.
Any knot equivalent to $U$ is also called an unknot, or a trivial knot.

## Remarks

The unknot bounds the flat unit disk, a [[Seifert surface|K866ND3WF7]] of genus $0$, and a knot is trivial exactly when it bounds an embedded disk.
Its [[knot group|AQVHM3FP0D]] is $\Z$, its [[crossing number|SWJZ4QRF9W]] is $0$, and its Alexander and Jones polynomials are $1$.
A tame knot with knot group $\Z$ is the unknot.
Recognizing the unknot from a diagram is the [[unknotting problem|VXSD46NRX0]]; Wolfgang Haken showed that it is decidable.
Some diagrams of the unknot are complicated, and untangling them may first require more crossings.

## Explanations

### L1 human

The unknot is a plain loop of string with no knot in it, like a rubber band.
Any loop that can be moved in space, without cutting, to a round circle is also an unknot.
It is the starting point of knot theory: to show that a knot is really knotted is to show that it is not the unknot.