Wild knots embedded in the Menger Sponge

TL;DR

This paper presents explicit recursive constructions of wild knots within the Menger sponge, controlling their wild points, and shows that certain dynamically defined wild knots can be isotoped into the sponge, emphasizing a constructive geometric approach.

Contribution

It introduces a constructive geometric method to embed and control wild knots in the Menger sponge, including those from Kleinian group actions.

Findings

Constructed infinitely many non-equivalent wild knots in the Menger sponge.

Controlled the wild points of knots within a Cantor set in the sponge.

Demonstrated isotopy of dynamically defined wild knots into the sponge.

Abstract

In this paper, we provide explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge, in such a way that we can control their set of wild points that lies in a usual Cantor set contained in the Menger sponge. Furthermore, we show that wild knots of dynamically defined type arising from Kleinian group actions can be isotoped into the sponge. We want to emphasize that our approach is constructive and geometric.