A knotty maths puzzle that dates back to the 1930s has finally been settled, and the answer came from the person who helped make the question famous.

In 2026, Columbia University mathematician Joan Birman, now 99, closed the case with University of Glasgow mathematician Tara Brendle and Princeton University mathematician Vasudha Bharathram. The trio showed that the Burau representation for four-strand braids is faithful.

Werner Burau, a German mathematician, introduced the problem in the 1930s. Mathematicians had already shown that knots could be reformulated as braids, made by dangling strands vertically and weaving them downward. Burau translated braid structures into algebraic objects called matrices, grids of numbers. The question was simple to state and hard to answer: did some of those matrices represent more than one braid?

If they did, the representation was called unfaithful.

Mathematicians had already pinned down most of the cases. For the first three strands of a braid, the matrices are faithful, with a one-to-one relationship. For five or more strands, the matrices are unfaithful. That left the four-strand braid as the only unresolved case.

“It’s pretty crazy,” says London Institute for Mathematical Sciences mathematician Yang-Hui He, who has worked on the problem. “It’s one of the most interesting stories in recent years, not just that there’s this proof but that so many people have worked on it over the last 70 years. It’s one of the major advances in the field of group and knot theory.”

Birman helped popularise the problem about 60 years ago. In the 1970s she wrote Braids, Links, and Mapping Class Groups, a foundational book for mathematicians. In it, she showed that the Burau representation for the four-strand group could be transformed into a search for relationships between special 3×3 matrices.

The result revived interest in the problem.

“Braids had been a backwater of topology,” Birman says. “Suddenly braids became very popular.”

A single-stranded braid is trivially faithful. The two-strand case, representing a repeating pattern of single crossovers, was quickly shown to be faithful. Around the same time Birman’s book came out, a pair of mathematicians showed that the three-strand group was also faithful.

Two decades later, a series of papers using a geometric approach first pioneered by mathematician John Moody showed that all braids with five or more strands were wholly unfaithful. Many researchers then became convinced the four-strand group must also be unfaithful and tried to find the missing relationship using Birman’s approach or Moody’s method.

He said he spent six months with Emmanuel Breuillard and Oleksandr Kosyak trying to get various artificial intelligence chatbots to work on Birman’s approach.

“Then, boom, on a Wednesday morning, Joan Birman herself with her two talented collaborators, claimed that they had solved the problem,” He says. “And they had.”

In the end, the 3×3 matrix approach did not lead to the proof. Nor did Moody’s method point to unfaithfulness. Bharathram instead suspected the assumption of unfaithfulness was wrong, so the group adapted Moody’s method to try to prove faithfulness.

Brendle described the proof using points and loops drawn on a piece of paper. The points correspond to strands, and the loops capture all possible interactions of those strands. The disks, defined as the insides of the loops, tell you how to calculate the Burau matrices.

“The more points you have, the more different loops you can draw,” Brendle says. “You can ask what different kinds of disks you can get.”

The trio found that as the disk variations increased, the ability to isolate a unique matrix weakened.

“Disks that look very different end up having a similar effect on the matrices,” Bharathram says, “meaning information is being lost in the translation from braids to matrices.”

That method let them test faithfulness across all possible braids. For three-strand braids, the possibilities were limited enough that the matrices remained faithful. Four-strand braids produced some difficult cases, but the trio handled them. With five or more strands, the disk possibilities multiplied enough to make the matrices unfaithful.

Brendle said the four-strand case marks a flash point for braids, comparing it to water undergoing a phase transition before freezing.

“I’m very surprised that it’s gotten this much attention,” Birman says, “There were many people who tried to prove this and it just didn’t work because they were looking for a counter-example to faithfulness.”

“It’s intrinsically interesting,” Bharathram says.

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