Companion Knot

Let K subset V subset S^3 be a knot that is geometrically essential in a standard embedding of the solid torus V in the three-sphere S^3. Let K_1 subset S^3 be another knot and let V_1 be a tubular neighborhood of K_1 in S^3. Finally, let h:V->V_1 be a homeomorphism and let K_2=h(K). Then K_1 is a companion knot of the knot K_2 (Rolfsen 1976, p. 111) and K_2 is called the satellite knot of K_1 (Adams 1994, pp. 115-118).

See also

Doubled Knot, Satellite Knot

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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 Knot

Cite this as:

Weisstein, Eric W. "Companion Knot." From MathWorld--A Wolfram Web Resource.

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