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In this paper we prove that a wild knot K which is the limit set of a Kleinian group acting conformally on the unit 3-sphere, with its standard metric, is homogeneous: given two points p, q ∈ K, there exists a homeomorphism f of the sphere such that f(K) = K and f(p) = q. We also show that if the wild knot is a fibered knot then we can choose an f which preserves the fibers.

Homogeneously wild curves and infinite knot products

H. Bothe (1981)

Fundamenta Mathematicae

Homotopy groups of arc complements in $S^{n}$ or Q

S. Singh (1988)

Colloquium Mathematicae

On the scarcity of tame disks in certain mild cells

Robert Daverman (1973)

Fundamenta Mathematicae

On the simultaneous embedding of uncountably many distinct wild arcs with one wild endpoint in $E^{3}$ , a geometric approach

Eugene Apodaca (1981)

Fundamenta Mathematicae

Open 3-manifolds, wild subsets of S3 and branched coverings.

José María Montesinos-Amilibia (2003)

Revista Matemática Complutense

In this paper, a representation of closed 3-manifolds as branched coverings of the 3-sphere, proved in [13], and showing a relationship between open 3-manifolds and wild knots and arcs will be illustrated by examples. It will be shown that there exist a 3-fold simple covering p : S3 --> S3 branched over the remarkable simple closed curve of Fox [4] (a wild knot). Moves are defined such that when applied to a branching set, the corresponding covering manifold remains unchanged, while the branching...

Representations of the Whitehead manifold $W h^{3}$ and Julia sets

Valentin Poénaru, Corrado Tanasi (1995)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Representing open 3-manifolds as 3-fold branched coverings.

José María Montesinos-Amilibia (2002)

Revista Matemática Complutense

It is proved that the Freudenthal compactification of an open, connected, oriented 3-manifold is a 3-fold branched covering of S3, and in some cases, a 2-fold branched covering of S3. The branching set is a locally finite disjoint union of strings.

Rigid sets in manifolds

David G. Wright (1986)

Banach Center Publications

Shrinking of toroidal decomposition spaces

Daniel Kasprowski, Mark Powell (2014)

Fundamenta Mathematicae

Given a sequence of oriented links L¹,L²,L³,... each of which has a distinguished, unknotted component, there is a decomposition space 𝓓 of S³ naturally associated to it, which is constructed as the components of the intersection of an infinite sequence of nested solid tori. The Bing and Whitehead continua are simple, well known examples. We give a necessary and sufficient criterion to determine whether 𝓓 is shrinkable, generalising previous work of F. Ancel and M. Starbird and others. This criterion...

Some New Invariants of Links.

Sadayoshi Kojima, Masyuki Yamasaki (1979)

Inventiones mathematicae

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