---
tag: A1E8TDEVJ9
kind: concept
name: Satellite knot
msc: [57K10]
specializes: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Satellite_knot
same_as:
  wikidata: Q7426084
  wikipedia: Satellite knot
---

## Summary

A satellite knot is a [[knot|QYEAZTYGPT]] that lies inside a thickened copy of another nontrivial knot, the companion, in an essential way.
Equivalently, its complement contains an incompressible torus that is not parallel to the boundary.
Satellite knots form one of the three classes in Thurston's division of knots, with [[torus knots|NS60VV786W]] and [[hyperbolic knots|K47G2Q24DT]].
Every [[knot sum|AGZSZW7GYE]] of two nontrivial knots is a satellite knot.

## Explanations

### L1 human

Take a nontrivial knot and thicken it into a knotted tube.
Now draw a new knot inside the tube, one that winds through the tube and cannot be pulled out of it or shrunk to a small ball inside it.
The new knot is a satellite, and the old knot is its companion.
Cables, where the new knot winds around the tube several times, are a simple example.
The granny knot, the sum of two trefoils, is a satellite with a trefoil as companion.