Dehn,s Lemma
المؤلف:
Hempel, J.
المصدر:
3-Manifolds. Princeton, NJ: Princeton University Press, 1976.
الجزء والصفحة:
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31-5-2021
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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.
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