Let 
be a knot that is geometrically essential
in a standard embedding of the solid torus 
in the three-sphere 
. Let 
be another knot and let 
be a tubular neighborhood
of 
in 
. Finally, let 
be a homeomorphism
and let 
.
Then 
is a companion knot of the knot 
(Rolfsen 1976, p. 111) and 
is called the satellite knot
of 
(Adams 1994, pp. 115-118).
Companion Knot
See also
Doubled Knot, Satellite KnotExplore with Wolfram|Alpha
References
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 115-118, 1994.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 225, 1976.Referenced on Wolfram|Alpha
Companion KnotCite this as:
Weisstein, Eric W. "Companion Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompanionKnot.html