Knot Complement

Let R^3 be the space in which a knot K sits. Then the space "around" the knot, i.e., everything but the knot itself, is denoted R^3-K and is called the knot complement of K (Adams 1994, p. 84).

If a knot complement is hyperbolic (in the sense that it admits a complete Riemannian metric of constant Gaussian curvature -1), then this metric is unique (Prasad 1973, Hoste et al. 1998).

See also

Complement, Compressible Surface, Knot, Knot Exterior

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Adams, C. C. "Knot Complements and Three-Manifolds." §9.1 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 243-246, 1994.Cipra, B. "To Have and Have Knot: When are Two Knots Alike?" Science 241, 1291-1292, 1988.Gordon, C. and Luecke, J. "Knots are Determined by their Complements." J. Amer. Math. Soc. 2, 371-415, 1989.Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1701936 Knots." Math. Intell. 20, 33-48, Fall 1998.Prasad, G. "Stong Rigidity of Q-Rank 1 Lattices." Invent. Math. 21, 255-286, 1973.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.

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Knot Complement

Cite this as:

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

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