Dehn's Lemma

An embedding of a 1-sphere in a 3-manifold which exists continuously over the 2-disk also extends over the disk as an embedding. An alternate phrasing is that if a knot group is isomorphic to the group of the integers , then the knot is isomorphic to the unknot (Livingston 1993, p. 104).

This theorem was proposed by Dehn in 1910, but a correct proof was not obtained until the work of Papakyriakopoulos (1957ab).

REFERENCES:

Hempel, J. 3-Manifolds. Princeton, NJ: Princeton University Press, 1976.

Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.

Papakyriakopoulos, C. D. "On Dehn's Lemma and the Asphericity of Knots." Proc. Nat. Acad. Sci. USA 43, 169-172, 1957a.

Papakyriakopoulos, C. D. "On Dehn's Lemma and the Asphericity of Knots." Ann. Math. 66, 1-26, 1957b.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 100-101, 1976.