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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0341494-9
PII: S 0002-9947(1974)0341494-9
Keywords: Wild knot, wild simple closed curve, homogeneous embedding, homogeneity
Article copyright: © Copyright 1974 American Mathematical Society