Homogeneity of dynamically defined wild knots

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 $p, q\in{K}$ 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.Comment: Accepted in Revista Matematica Complutens

Where the Wild Knots Are

The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a standard unknot whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require the aforementioned unknot with a countable sequence of crossings; building up the machinery to work with these kinds of objects is the focus of the second part of the project. Initially, the presence of infinitely-many crossing might appear to be a contradiction to the finiteness constraint in Reidemeister\u27s theorem; we show that this is not the case, and introduce the notion of feral points to represent areas of our diagrams in which it is not immediately obvious whether the knot is wild or tame. We employ uniform convergence to create sufficient conditions for guaranteeing ambient isotopy under limits and resolve a seeming contradiction given by the wild arc of Fox-Artin. Finally, we show that any knot whose crossings are topologically discrete is ambient isotopic to a countable union of polygonal segments, and discuss implications for extending Reidemeister\u27s theorem in this context

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

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 set changes and becomes wild. As a consequence every closed, oriented 3-manifold is represented as a 3-fold covering of S3 branched over a wild knot, in plenty of different ways, confirming the versatility of irregular branched coverings. Other collection of examples is obtained by pasting the members of an infinite sequence of two-component strongly-invertible link exteriors. These open 3-manifolds are shown to be 2-fold branched coverings of wild knots in the 3-sphere Two concrete examples, are studied: the solenoidal manifold, and the Whitehead manifold. Both are 2-fold covering of the euclidean space R3 branched over an uncountable collection of string projections in R3.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 : S 3 S 3 branched over the re markable simple closed curve of Fox [4] (a wild knot). Moves are defined such that when applied to a branching set, the corresponding covering manifold re mains unchanged, while the branching set changes and becomes wild. As a consequence every closed, oriented 3-manifold is represented as a 3-fold cov ering of S 3 branched over a wild knot, in plenty of different ways, confirming the versatility of irregular branched coverings. Other collection of examples is obtained by pasting the members of an infinite sequence of two-component strongly-invertible link exteriors. These open 3-manifolds are shown to be 2 fold branched coverings of wild knots in the 3-sphere Two concrete examples, are studied: the solenoidal manifold, and the Whitehead manifold. Both are 2 fold covering of the euclidean space R3 branched over an uncountable collection of string pro jections in R3 . 2000 Mathematics Sub ject Classification: 57M12, 57M30, 57N10