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

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.