The duo kept their program running in the background for over a decade. During that time, a couple of computers from their ragtag collection succumbed to overheating and even flames. “There was one that actually sent out sparks,” Brittenham said. “That was kind of fun.” (Those 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 Brittenham and Hermiller’s attention. Perhaps, they thought, machine learning wasn’t the best approach for this particular problem: If a counterexample to the additivity conjecture was out there, it would be “a needle in a haystack,” Hermiller said. “That’s not quite what things like machine learning are about. They’re about trying to find patterns in things.”
But it reinforced a suspicion the pair already had — that maybe their more carefully honed sneakernet could find the needle.
The Tie That Binds
Brittenham and Hermiller realized they could make use of the unknotting sequences they’d uncovered to look for potential counterexamples to the additivity conjecture.
Imagine again that you have two knots whose unknotting numbers are 2 and 3, and you’re trying to unknot their connect sum. After one crossing change, you get a new knot. If the additivity conjecture is to be believed, then the original knot’s unknotting number should be 5, and this new knot’s should be 4.
But what if this new knot’s unknotting number is already known to be 3? That implies that the original knot can be untied in just four steps, breaking the conjecture.
“We get these middle knots,” Brittenham said. “What can we learn from them?”
He and Hermiller already had the perfect tool for the occasion humming away on their suite of laptops: the database they’d spent the previous decade developing, with its upper bounds on the unknotting numbers of thousands of knots.
The mathematicians started to add pairs of knots and work through the unknotting sequences of their connect sums. They focused on connect sums whose unknotting numbers had only been approximated in the loosest sense, with a big gap between their highest and lowest possible values. But that still left them with a massive list of knots to work through — “definitely in the tens of millions, and probably in the hundreds of millions,” Brittenham said.
For months, their computer program applied crossing changes to these knots and compared the resulting knots 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 turned up. To his great surprise, there was a line of text: “CONNECT SUM BROKEN.” It was a message he and Hermiller had coded into the program — but they’d never expected to actually see it.
Initially, they were doubtful of the result. “The very first thing that went through our heads was there was something wrong with our programming,” Brittenham said.
“We just dropped absolutely everything else,” Hermiller recalled. “All of life just went away. Eating, sleeping got annoying.”
But their program checked out. They even tied the knot it had identified in a rope, then worked through the unknotting procedure by hand, just to make sure.
Their counterexample was real.
Twisted Mysteries
The counterexample Brittenham and Hermiller found is built out of two copies of a knot called the (2, 7) torus knot. This knot is made by winding two strings around each other three and a half times and then gluing their opposing ends together. Its mirror image is made by winding three and a half times in the other direction.
The unknotting number of both the (2, 7) torus knot and its mirror image is 3. But Brittenham and Hermiller’s program found that if you add these knots, you can unknot the result in just five steps — not six, as the additivity conjecture predicted.