Complex knots can actually be easier to untie than simple ones

Mathematicians have solved a decades-old problem in knot theory, discovering that linking two knots together can actually produce a knot that is easier to untie – the opposite of what was expected

By Matthew Sparkes

15 July 2025

A knotty problem for mathematicians finally has a solution

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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. “

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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.