Schönflies Theorem

If  is a simple closed curve in , the closure of one of the components of  is homeomorphic with the unit 2-ball. This theorem may be proved using the Riemann mapping theorem, but the easiest proof is via Morse theory.

The generalization to  dimensions is called Mazur's theorem. It follows from the Schönflies theorem that any two knots of  in  or  are equivalent.

REFERENCES:

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 9, 1976.

Thomassen, C. "The Jordan-Schönflies Theorem and the Classification of Surfaces." Amer. Math. Monthly 99, 116-130, 1992.