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.
When the paper was posted, I gasped out loud. Allison Moore
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