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Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal [quantamagazine.org]:

Menger’s statement didn’t distinguish between homeomorphic curves. His proof only guaranteed, for instance, that the circle could be found in his sponge — not that all homeomorphic knots could be, their loops and tangles still intact. Malors wanted to prove that you could find every knot within the sponge.

It seemed like the right mashup to excite young mathematicians. They’d recently had fun learning about knots in his seminar. And who doesn’t love a fractal? The question was whether the problem would be approachable. “I really hoped there was an answer,” Malors said.

There was. After just a few months of weekly Zoom meetings with Malors, three of his high school students — Joshua Broden, Noah Nazareth and Niko Voth — were able to show that all knots can indeed be found inside the Menger sponge [arxiv.org]. Moreover, they found that the same can likely be said of another related fractal, too.

“It’s a clever way of putting things together,” said Radmila Sazdanovic, [ncsu.edu] a topologist at North Carolina State University who was not involved in the work. In revisiting Menger’s century-old theorem, she added, Malors — who usually does research in the disparate field of number theory — had apparently asked a question that no one thought to ask before. “This is a very, very original idea,” she said.

A Different Way to See Knots

Broden, Nazareth and Voth had taken several of Malors’ summer workshops over the years. When he first taught them about knots in an earlier workshop, “it blew 14-year-old me’s mind,” said Voth.

But the Menger problem would be their first time moving beyond school workbooks with answer keys. “It was a little bit nerve-racking, because it was the first time I was doing something where truly nobody has the answer, not even Malors,” said Nazareth. Maybe there was no answer at all.

Their goal was essentially to thread a microscopic sewing needle through a cloud of dust — the material that remained of the sponge after many removals. They would have to stick the pin in the right places, tie the knotted tangles with immaculate precision, and never leave the sponge. If their thread ended up floating in the empty holes of the sponge for any knot, it was game over.

Not an easy task. But there was a way to simplify it. Knots can be depicted on a flat piece of paper as special diagrams called arc presentations. To create one, you start with information about how the strands of your knot pass in front of or behind each other. Then you apply a set of rules to translate this information into a series of points on a grid. Every row and column of the grid will contain exactly two points.

Original Submission