Knots Inside Fractals

2024/09/05 by Joshua Broden, Broden, Joshua, Malors Espinosa +5 · 4 voices · 1 citation
Mathematics · #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #math.GN #math.GT

paper · pdf · doi:10.48550/arxiv.2409.03639

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.

Cited by

• Any smooth n-dimensional knot \mathbbSⁿ\hookrightarrowℝⁿ⁺² is isotopic to an n-knot contained in the Menger Mⁿ⁺²_n-continuum

Discussions

• Their paper is pretty short and only requires advanced undergrad math to comprehend: arxiv.org/pdf/2409.03639 Their proof is also visually very intuitive [bsky, 2 points, 2 comments]
• Wonderful new paper on embedding in knots arxiv.org/abs/2409.03639 [bsky, 0 points, 0 comments]
• Now find the transformation in the sponge arxiv.org/abs/2409.03639 #reference #fractal #knot #math [bsky, 0 points, 0 comments]
• Cool work, and done by some bright teenage mathematicians arxiv.org/abs/2409.03639 [bsky, 0 points, 0 comments]

Related

• Plumbers' knots
• Fibered ribbon pretzels
• The representativity of pretzel knots
• Under-knotted and over-knotted polymers: compact self-avoiding loops
• Non-slice 3-stranded pretzel knots