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).
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. USA43, 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.
, then the knot is isomorphic to the
unknot (Livingston 1993, p. 104).
