$N$-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group

TL;DR

This paper constructs higher-dimensional wild knots with Cantor sets of wild points using conformal Schottky groups, generalizing previous results and showing that fibered properties are preserved.

Contribution

It introduces a method to create wild knots from beaded necklaces and Schottky groups, extending known constructions to higher dimensions and analyzing their fibered and covering properties.

Findings

Wild knots with Cantor sets of wild points are constructed.

The wild knots preserve the fibered property of the original knots.

Cyclic branched coverings of the wild knots are studied.

Abstract

Starting with a smooth, non-trivial $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on $\bS^{n+2}$, generated by inversions on the spheres which are the boundary of the ``beads''. We show that if $K$ is a fibered knot, then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots. This work generalizes the result presented in [8].