page_title: Satellite knot
section_title: Examples
hierarchical_section_title: Satellite knot / Examples
caption_attribution_description: English: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.
context_page_description: In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion.
  A satellite knot can be picturesquely described as follows: start by taking a nontrivial knot lying inside an unknotted solid torus. Here "nontrivial" means that the knot is not allowed to sit inside of a 3-ball in and is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot.
  This means there is a non-trivial embedding and. The central core curve of the solid torus is sent to a knot, which is called the "companion knot" and is thought of as the planet around which the "satellite knot" orbits. The construction ensures that is a non-boundary parallel incompressible torus in the complement of.
context_section_description: Example 1: The connect-sum of a figure-8 knot and trefoil.
  Example 2: Untwisted Whitehead double of a figure-8.
  Example 3: Cable of a connect-sum.
  Example 4: Cable of trefoil.
  Examples 5 and 6 are variants on the same construction. They both have two non-parallel, non-boundary-parallel incompressible tori in their complements, splitting the complement into the union of three manifolds. In Example 5 those manifolds are: the Borromean rings complement, trefoil complement and figure-8 complement. In Example 6 the figure-8 complement is replaced by another trefoil complement.
caption: Example 5: A knot which is a 2-fold satellite i.e.: it has non-parallel swallow-follow tori.
exif: {}