---
tag: C01XBGAHC6
kind: example
name: Figure-eight knot
msc: [57K10, 57K32]
instance_of: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
same_as:
  wikidata: Q168697
  wikipedia: Figure-eight knot (mathematics)
---

## Example

The *figure-eight knot* is the [[knot|QYEAZTYGPT]] obtained by closing up the figure-eight knot of rope.
Its standard diagram has $4$ crossings that alternate over and under along the curve.
One parametrization of it is
$$t \mapsto \bigl((2 + \cos 2t) \cos 3t, \ (2 + \cos 2t) \sin 3t, \ \sin 4t\bigr), \qquad t \in [0, 2\pi).$$
Its [[crossing number|SWJZ4QRF9W]] is $4$.
Indeed, the only knots with crossing number less than $4$ are the unknot and the two trefoils, and the figure-eight knot is neither.

## Remarks

Its Alexander polynomial is $-t^{-1} + 3 - t$, which differs from that of the unknot, $1$, and from that of the trefoil, $t^{-1} - 1 + t$.
It is not [[tricolorable|KNAGK0W65G]], because its determinant is $5$, not divisible by $3$.
It is equivalent to its mirror image, unlike the trefoil.
It is a [[hyperbolic knot|K47G2Q24DT]]: its complement carries a complete hyperbolic metric of finite volume, made of two regular ideal tetrahedra.

## Explanations

### L1 human

The figure-eight knot is the next simplest knot after the trefoil, with four crossings in its simplest picture.
Its name comes from the figure-eight knot that climbers and sailors tie, with the ends joined.
Unlike the trefoil, it can be deformed into its own mirror image.