Homogeneity and extension properties of embeddings of in

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 are explored in this paper. Let be a simple closed curve and an embedding of in . 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 such that and . The embedding *f* or the simple closed curve *S* is *extendible* if any automorphism of *S* extends to an automorphism of . 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.

**[1]**Ralph H. Fox and Emil Artin,*Some wild cells and spheres in three-dimensional space*, Ann. of Math. (2)**49**(1948), 979–990. MR**0027512****[2]**R. H. Bing,*A simple closed curve that pierces no disk*, J. Math. Pures Appl. (9)**35**(1956), 337–343. MR**0081461****[3]**H. G. Bothe,*Ein homogen wilder Knoten*, Fund. Math.**60**(1967), 271–283 (German). MR**0216494****[4]**H. G. Bothe,*Eine fixierte Kurve in 𝐸³*, General Topology and its Relations to Modern Analysis and Algebra, II (Proc. Second Prague Topological Sympos., 1966) Academia, Prague, 1967, pp. 68–73 (German). MR**0236898****[5]**C. H. Edwards Jr.,*Concentric solid tori in the 3-sphere*, Trans. Amer. Math. Soc.**102**(1962), 1–17. MR**0140091**, 10.1090/S0002-9947-1962-0140091-X**[6]**R. H. Fox,*A quick trip through knot theory*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR**0140099****[7]**Horst Schubert,*Knoten und Vollringe*, Acta Math.**90**(1953), 131–286 (German). MR**0072482****[8]**Arnold C. Shilepsky,*Homogeneity by isotopy for simple closed curves*, Duke Math. J.**40**(1973), 463–472. MR**0319179**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0341494-9

Keywords:
Wild knot,
wild simple closed curve,
homogeneous embedding,
homogeneity

Article copyright:
© Copyright 1974
American Mathematical Society