Two distinct knots cannot have the same exterior. Or, equivalently, a knot is completely determined by its knot exterior (Cipra 1988; Adams 1994, p. 261). The question was first posed by Tietze in 1908 and finally proved by Gordon and Luecke (1989).
Gordon-Luecke Theorem
See also
Knot ExteriorExplore with Wolfram|Alpha
References
Adams, C. C. "The Poincaré Conjecture, Dehn Surgery, and the Gordon-Luecke Theorem." §9.3 in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 257-263, 1994.Cipra, B. A. "To Have and Have Knot: When Are 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.Referenced on Wolfram|Alpha
Gordon-Luecke TheoremCite this as:
Weisstein, Eric W. "Gordon-Luecke Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Gordon-LueckeTheorem.html