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 Z, 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).

See also

Knot Group

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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.

Referenced on Wolfram|Alpha

Dehn's Lemma

Cite this as:

Weisstein, Eric W. "Dehn's Lemma." From MathWorld--A Wolfram Web Resource.

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