Imagine again that you have two knots whose unknotting numbers are 2 and 3, and you’re trying to unknot their connect sum. 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? “We get these middle knots,” Brittenham said. For months, their computer program applied crossing changes to these knots and compared the resulting knots to those in their database.