Unpicking How to Measure the Complexity of Knots

The duo kept their program in the background for over a decade. Meanwhile, a few computers in their motley collection succumbed to overheating and even flames. “There was one that was sending sparks,” Brittenham said. “It was pretty fun.” (These machines, he added, were “honorably retired.”)

Then, in the fall of 2024, a paper about a failed attempt to use machine learning to disprove the additivity conjecture caught the attention of Brittenham and Hermiller. Perhaps, they thought, machine learning was not the best approach for this particular problem: If there was a counterexample to the additivity conjecture, it would be “a needle in a haystack,” Hermiller said. “That’s not really the point of things like machine learning. It’s about trying to find patterns in things.”

But it reinforced suspicions both men already had: Maybe their sneaker network, more carefully honed, could find the needle.

The link that binds

Brittenham and Hermiller realized that they could use the denouement sequences they had discovered to look for potential counterexamples to the additivity conjecture.

Imagine again that you have two nodes whose unknot numbers are 2 and 3, and you are trying to unknot their connection sum. After a crossover change you get a new node. If we believe the additivity conjecture, then the number of unraveling of the original node should be 5, and that of this new node should be 4.

But what happens if the unwind number of this new node is already known to be 3? This implies that the initial knot can be untied in just four steps, breaking the guesswork.

“We get these nodes in the middle,” Brittenham said. “What can we learn from them?

He and Hermiller already had the perfect tool for the occasion, humming on their suite of laptops: the database they had spent the previous decade developing, with its upper limits on the number of thousands of knots to unravel.

Mathematicians began adding pairs of nodes and unraveling the sequences of their connection sums. They focused on connection sums whose settlement numbers had only been approximated in the broadest sense, with a large gap between their highest and lowest possible values. But that still left them with a huge list of knots to resolve — “certainly in the tens of millions, and probably in the hundreds of millions,” Brittenham said.

For months, their computer program applied crossover edits to these nodes and compared the resulting nodes to those in their database. One day in late spring, Brittenham checked the program’s output files, as he did most days, to see if anything interesting had appeared. To his surprise, there was a line of text: “CONNECT SUM BROKEN.” It was a message that he and Hermiller had coded into the program, but they never expected to actually see it.