The duo saved 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 mentioned. “That was sort of enjoyable.” (These machines, he added, have been “honorably retired.”)
Then, within the fall of 2024, a paper a few failed try to make use of machine studying to disprove the additivity conjecture caught Brittenham and Hermiller’s consideration. Maybe, they thought, machine studying wasn’t the very best method for this specific drawback: If a counterexample to the additivity conjecture was on the market, it might be “a needle in a haystack,” Hermiller mentioned. “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 perhaps their extra rigorously honed sneakernet might 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 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 must be 5, and this new knot’s must be 4.
However what if this new knot’s unknotting quantity is already identified to be 3? That means that the unique knot could be untied in simply 4 steps, breaking the conjecture.
“We get these center knots,” Brittenham mentioned. “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 1000’s of knots.
The mathematicians began so as to add pairs of knots and work by means of the unknotting sequences of their join sums. They centered on join sums whose unknotting numbers had solely been approximated within the loosest sense, with a giant hole between their highest and lowest potential values. However that also left them with an enormous record of knots to work by means of—“positively within the tens of tens of millions, and doubtless within the a whole lot of tens of millions,” Brittenham mentioned.
For months, their pc program utilized crossing adjustments 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 fascinating 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.