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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Homogeneity and extension properties of embeddings of $ S\sp{1}$ in $ E\sp{3}$

Author: Arnold C. Shilepsky
Journal: Trans. Amer. Math. Soc. 195 (1974), 265-276
MSC: Primary 57A10; Secondary 55A30
MathSciNet review: 0341494
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Abstract: Two properties of embeddings of simple closed curves in $ {E^3}$ are explored in this paper. Let $ {S^1}$ be a simple closed curve and $ f({S^1}) = S$ an embedding of $ {S^1}$ in $ {E^3}$. The simple closed curve S is homogeneously embedded or alternatively f is homogeneous if for any points p and q of S, there is an automorphism h of $ {E^3}$ such that $ h(S) = S$ and $ h(p) = q$. The embedding f or the simple closed curve S is extendible if any automorphism of S extends to an automorphism of $ {E^3}$. Two classes of wild simple closed curves are constructed and are shown to be homogeneously embedded. A new example of an extendible simple closed curve is constructed. A theorem of H. G. Bothe about extending orientation-preserving automorphisms of a simple closed curve is generalized.

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  • [1] R. H. Fox and E. Artin, Some wild cells and spheres in three dimensional space, Ann. of Math. (2) 49 (1948), 979-990. MR 10, 317. MR 0027512 (10:317g)
  • [2] R. H. Bing, A simple closed curve that pierces no disk, J. Math. Pures Appl. (9) 35 (1956), 337-343. MR 18, 407. MR 0081461 (18:407b)
  • [3] H. G. Bothe, Ein homogen wilder Knoten, Fund. Math. 60 (1967), 271-283. MR 35 #7327. MR 0216494 (35:7327)
  • [4] H. G. Bothe, Eine fixierte Kurve in $ {E^3}$, General Topology and its Relations to Modern Analysis and Algebra, II (Proc. Second Prague Topological Sympos., 1966), Academia, Prague, 1967, pp. 68-73. MR 38 #5191. MR 0236898 (38:5191)
  • [5] C. H. Edwards, Concentric solid tori in the 3-spheres, Trans. Amer. Math. Soc. 102 (1962), 1-17. MR 25 #3514. MR 0140091 (25:3514)
  • [6] R. H. Fox, Quick trip through knot theory, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 120-167. MR 25 #3522. MR 0140099 (25:3522)
  • [7] H. Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131-286. MR 17, 291. MR 0072482 (17:291d)
  • [8] A. C. Shilepsky, Homogeneity by isotopy for simple closed curves, Duke Math. J. 40 (1973), 63-72. MR 0319179 (47:7725)

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Additional Information

PII: S 0002-9947(1974)0341494-9
Keywords: Wild knot, wild simple closed curve, homogeneous embedding, homogeneity
Article copyright: © Copyright 1974 American Mathematical Society