Existence of iterative fractals that admit some knot families but exclude others

Investigate whether there exists a fractal produced by an iterative process that admits embeddings of certain knot types while excluding others, and ascertain what such selectivity would imply about the fractal’s structure or the avoided knot families.

Background

After demonstrating that all knots can be embedded in the Menger Sponge and many (including all pretzel knots) appear in the Sierpinski Tetrahedron, the authors pose a broader structural question about fractals generated iteratively: can some fractals inherently admit only certain knot families?

A positive example would provide new insights into the relationship between fractal geometry and knot type constraints, potentially yielding classification results or complexity measures tied to iterative fractal constructions.