Doubled Knot

Let  be the knot above, and let the homomorphism  taking a knot  to its companion knot  be faithful (i.e., taking the preferred longitude and meridian of the original torus to the referred longitude and latitude of the tubular neighborhood). Then  is an untwisted doubled knot of . Otherwise,  is a doubled knot with  twists (Rolfsen 1976, p. 112).

REFERENCES:

Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., pp. 35, 73, 146, and 154-155, 1993.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 112, 166-167, and 225, 1976.

Whitehead, J. H. C. "On Doubled Knots." J. London Math. Soc. 12, 63-71, 1937.