Complex knots can actually be easier to untie than simple ones

Why is untangling two small knots more difficult than unravelling one big one? Surprisingly, mathematicians have found that larger and seemingly more complex knots created by joining two simpler ones together can sometimes be easier to undo, invalidating a conjecture posed almost 90 years ago.

“We were looking for a counterexample without really having an expectation of finding one, because this conjecture had been around so long,” says Mark Brittenham at the University of Nebraska at Lincoln. “In the back of our heads, we were thinking that the conjecture was likely to be true. It was very unexpected and very surprising. “

Mathematicians like Brittenham study knots by treating them as tangled loops with joined ends. One of the most important concepts in knot theory is that each knot has an unknotting number, which is the number of times you would have to sever the string, move another piece of the loop through the gap and then re-join the ends before you reached a circle with no crossings at all – known as the “unknot”.

Calculating unknotting numbers can be a very computationally intensive task, and there are still knots with as few as 10 crossings that have no solution. Because of this, it can be helpful to break knots down into two or more simpler knots to analyse them, with those that can’t be split any further known as prime knots, analogous to prime numbers.

But a long-standing mystery is whether the unknotting numbers of the two knots added together would give you the unknotting number of the larger knot. Intuitively, it might make sense that a combined knot would be at least as hard to undo as the sum of its constituent parts, and in 1937, it was conjectured that undoing the combined knot could never be easier.

Now, Brittenham and Susan Hermiller, also at the University of Nebraska at Lincoln, have shown that there are cases when this isn’t true. “The conjecture’s been around for 88 years and as people continue not to find anything wrong with it, people get more hopeful that it’s true,” says Hermiller. “First, we found one, and then quickly we found infinitely many pairs of knots for whom the connected sum had unknotting numbers that were strictly less than the sum of the unknotting numbers of the two pieces.”

“We’ve shown that we don’t understand unknotting numbers nearly as well as we thought we did,” says Brittenham. “There could be – even for knots that aren’t connected sums – more efficient ways than we ever imagined for unknotting them. Our hope is that this has really opened up a new door for researchers to start exploring.”

While finding and checking the counterexamples involved a combination of existing knowledge, intuition and computing power, the final stage of checking the proof was done in a decidedly more simple and practical manner: tying the knot with a piece of rope and physically untangling it to show that the researchers’ predicted unknotting number was correct.

Andras Juhasz at the University of Oxford, who previously worked with AI company DeepMind to prove a different conjecture in knot theory, says that he and the company had tried unsuccessfully to crack this latest problem about additive sets in the same way, but with no luck.

“We spent at least a year or two trying to find a counterexample and without success, so we gave up,” says Juhasz. “It is possible that for finding counterexamples that are like a needle in a haystack, AI is maybe not the best tool. This was a hard-to-find counterexample, I believe, because we searched pretty hard.”

Despite there being many practical applications for knot theory, from cryptography to molecular biology, Nicholas Jackson at the University of Warwick, UK, is hesitant to suggest that this new result can be put to good use. “I guess we now understand a little bit more about how circles work in three dimensions than we did before,” he says. “A thing that we didn’t understand quite so well a couple of months ago is now understood slightly better.”