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Wild knots embedded in the Menger Sponge

Gabriela Hinojosa, Ulises Morales-Fuentes, Rogelio Valdez, Alberto Verjovsky

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.

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.
Paper Structure (9 sections, 7 theorems, 21 equations, 12 figures)

This paper contains 9 sections, 7 theorems, 21 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Menger Sponge
  4. Knots
  5. Wild knots embedded in the Menger sponge
  6. Squareflake curves
  7. Main theorem
  8. Wild knots of dynamically defined type in the Menger sponge
  9. Wild knots embedded in the 2-dimensional Sierpiński carpet ${\mathcal{S}}^2$

Key Result

Lemma 2.1

If $(x,y)$ is a point in the Cantor dust, then for all $0\leq z\leq 1$ the point $(x,y,z)$ belongs to the Menger sponge $M$.

Figures (12)

  • Figure 1: First steps in the construction of the Menger sponge.
  • Figure 2: A tame knot and its polygonal representative.
  • Figure 3: A locally tame point
  • Figure 4: The figure-eight knot on the Menger sponge.
  • Figure 5: The curve $S$.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.4
  • Lemma 3.5
  • Remark 3.6
  • ...and 2 more