In the 1930s, Werner Burau, a German mathematician, introduced a twisting geometric mystery that would last nearly a century.
Previously, mathematicians had shown that knots could be rephrased into something more relevant: braids. A “braid” begins with a collection of strands. To make the braid, hang the strands vertically and weave them downward as desired. Any type of knot, no matter how complicated, can be made into a braid
As part of his investigation, Burau carefully translated braid structures into algebraic objects, making them much easier to manipulate mathematically. The objects, called matrices, are grids of numbers that work much like a spreadsheet. But mathematicians at the time worried that his elegant translation might lose information. Did any of these dies represent more than one braid? If this was the case, they were called “infidels.” The problem was determining which, if any, of her braid depictions were unfaithful.
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Columbia University mathematician Joan Birman helped popularize the question about 60 years ago. And in 2026, nearly a century after the Burau representation was first proposed, Birman, now 99, closed the case with the help of two collaborators, University of Glasgow mathematician Tara Brendle and Princeton University mathematician Vasudha Bharathram. We have known for a long time that for the first three strands of a braid, the matrices are always faithful, that is to say that there is a one-to-one relationship between them. We also knew that for five or more strands, the matrices all become unfaithful. But in a new twist, mathematicians have shown how the four-strand braid complicates the relationship, but ultimately its matrices remain faithful.
“It’s pretty crazy,” says Yang-Hui He, a mathematician at the Institute of Mathematical Sciences in London, who has worked on the problem. “This is one of the most interesting stories in recent years, not only because there is proof, but also because so many people have worked on it over the last 70 years. It is one of the major advances in the field of group and knot theory.”
As a girl, Birman shared an interest with her grandmother: knitting. It was only when she arrived at higher education that she discovered the beautiful mathematical side to his hobby. Although she says the implications of her mathematical work for her hobby are exaggerated (Bharathram points out the major hurdle of translating a sweater into a matrix), the subject became Burman’s lifelong obsession.
In the 1970s, Birman wrote a fundamental book for mathematicians: Braids, Links, and Mapping Class Groups. She shows that Burau’s representation for the four-strand group can be transformed into a search for relationships between special three-by-three (3 × 3) matrices. The discovery reopened Burau’s cold case. “The braids were a lost corner of the topology,” explains Birman. “All of a sudden, braids became very popular.”
A single strand braid is trivially faithful. Likewise, the argument for two strands, representing a repeating pattern of simple crossovers, quickly proved to be true. Around the same time that Birman’s book appeared, two mathematicians demonstrated that the three-pronged group was also loyal. But two decades later, a surprise awaited him. A series of papers using a geometric approach first pioneered by mathematician John Moody, showed that all braids with five or more strands were entirely Andfaithful.
This left the case of four-strand braids as the only holdout. Many researchers were convinced that the group must also be unfaithful and sought to find the relationship in Birman’s approach or in Moody’s method.
The race was on.
“I was introduced to the problem,” he says, “by Emmanuel Breuillard and Sasha. [Oleksandr] Kosyak,” two well-known mathematicians who have been working on this dilemma for more than 20 years. The group spent six fruitless months trying to pilot various artificial intelligence chatbots using Birman’s approach. “Then, boom, one Wednesday morning, Joan Birman herself and her two talented collaborators claimed that they had solved the problem,” he says. “And they did.”
Ultimately, the 3×3 matrix approach had been a false lead. Just like Moody’s infidelity angle. Bharatram thought the infidelity hypothesis might be false, so the group adapted Moody’s method to try to prove fidelity. His intuition put the trio on the right path.
Brendle offered an explanation of the group’s proof: Take a piece of paper and draw a bunch of dots. Then draw loops around some of these points. Dots correspond to strands, while loops capture all possible interactions of those strands. The disks, which are defined as the inside of the loops, tell you how to calculate Burau matrices. “The more points you have, the more different loops you can draw,” says Brendle. “You can ask what different types of discs you can get.”
The trio discovered that as disc variations increased, the ability to isolate a unique weakened matrix. “Disks that look very different end up having a similar effect on the matrices,” explains Bharathram, “meaning that information is lost when translating the braids into matrices.”
The method has proven effective in testing the fidelity of all possible braids. For three-strand braids, the possibilities were so limited that dies had no choice but to be faithful. The four-strand braids presented a few unpleasant scenarios, but the trio was able to handle them. However, when braids had five or more strands, disc possibilities proliferated, ensuring that the dies were all unfaithful. Just as water undergoes a phase transition before freezing, Brendle says, the four-strand casing represents an important flash point for braids.
“I’m very surprised that it’s getting so much attention,” Birman says. “A lot of people tried to prove it and it just didn’t work because they were looking for a counterexample of faithfulness.”
Bharathram, Burman and Brendle, scattered in different cities and continents, did not celebrate after completing their proof. Although braids are making their way into many different areas of science, from protein folding to string theory, simply solving the century-old problem was reward enough for the trio. “It’s inherently interesting,” Bharathram says.
