Companion Knot

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).

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.