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

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

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?" Science241, 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 Knots." Math. Intell.20, 33-48,
Fall 1998.Prasad, G. "Stong Rigidity of -Rank 1 Lattices." Invent. Math.21, 255-286,
1973.Rolfsen, D. Knots
and Links. Wilmington, DE: Publish or Perish Press, 1976.