Wild knots embedded in the Menger Sponge

Abstract: In this paper, we prove that there exist infinitely many non-equivalent wild knots embedded in the Menger sponge, each with its set of wild points being a subset of a tame Cantor set. Even more, any wild knot of dynamically defined type obtained as the inverse limit of a sequence of nested pearl chain necklaces constructed via the action of a Kleinian group is isotopic to a wild knot contained in the Menger sponge. As a consequence, there are infinitely many wild knots contained in the Sierpiński 2-dimensional carpet.