Knots Inside Fractals

TL;DR

This paper demonstrates that all knots can be embedded into certain fractals, specifically the Menger Sponge and Sierpinski Tetrahedron, and compares their complexity based on the iterations needed for embedding.

Contribution

It proves that all knots can be embedded into the Menger Sponge and all Pretzel knots into the Sierpinski Tetrahedron, establishing a new connection between knot theory and fractals.

Findings

All knots can be embedded into the Menger Sponge.

All Pretzel knots can be embedded into the Sierpinski Tetrahedron.

The complexity of fractals can be compared by the number of iterations needed for knot embedding.

Abstract

We prove that all knots can be embedded into the Menger Sponge fractal. We prove that all Pretzel knots can be embedded into the Sierpinski Tetrahedron. Then we compare the number of iterations of each of these fractals needed to produce a given knot as a mean to compare the complexity of the two fractals.