The duo stored their program working within the background for over a decade. Throughout that point, a few computer systems from their ragtag assortment succumbed to overheating and even flames. “There was one that really despatched out sparks,” Brittenham stated. “That was type of enjoyable.” (These machines, he added, had been “honorably retired.”)
Then, within the fall of 2024, a paper a few failed attempt to use machine learning to disprove the additivity conjecture caught Brittenham and Hermiller’s consideration. Maybe, they thought, machine studying wasn’t the perfect method for this specific downside: If a counterexample to the additivity conjecture was on the market, it could be “a needle in a haystack,” Hermiller stated. “That’s not fairly what issues like machine studying are about. They’re about looking for patterns in issues.”
But it surely strengthened a suspicion the pair already had—that possibly their extra fastidiously honed sneakernet may discover the needle.
The Tie That Binds
Brittenham and Hermiller realized they may make use of the unknotting sequences they’d uncovered to search for potential counterexamples to the additivity conjecture.
Think about once more that you’ve got two knots whose unknotting numbers are 2 and three, and also you’re making an attempt to unknot their join sum. After one crossing change, you get a brand new knot. If the additivity conjecture is to be believed, then the unique knot’s unknotting quantity ought to be 5, and this new knot’s ought to be 4.
However what if this new knot’s unknotting quantity is already identified to be 3? That means that the unique knot might be untied in simply 4 steps, breaking the conjecture.
“We get these center knots,” Brittenham stated. “What can we be taught from them?”
He and Hermiller already had the right device for the event buzzing away on their suite of laptops: the database they’d spent the earlier decade creating, with its higher bounds on the unknotting numbers of hundreds of knots.
The mathematicians began so as to add pairs of knots and work via the unknotting sequences of their join sums. They centered on join sums whose unknotting numbers had solely been approximated within the loosest sense, with an enormous hole between their highest and lowest doable values. However that also left them with a large listing of knots to work via—“undoubtedly within the tens of thousands and thousands, and doubtless within the a whole bunch of thousands and thousands,” Brittenham stated.
For months, their pc program utilized crossing modifications to those knots and in contrast the ensuing knots to these of their database. Sooner or later in late spring, Brittenham checked this system’s output recordsdata, as he did most days, to see if something attention-grabbing had turned up. To his nice shock, there was a line of textual content: “CONNECT SUM BROKEN.” It was a message he and Hermiller had coded into this system—however they’d by no means anticipated to really see it.
