Ramanujan's Century-Old Equations Turn Out to Be Knots — and There Are Infinitely Many
Some maths problems sit quietly for a hundred years and then turn out to be something else entirely. That has just happened to two of the most famous formulas in mathematics — equations written down by the Indian mathematician Srinivasa Ramanujan, who died in 1920 at the age of 32.
The formulas are called the Rogers–Ramanujan identities. Each one says something surprising: two wildly different-looking infinite sums are actually equal — and equal, moreover, to a simple infinite product. One side counts ways of splitting whole numbers into pieces (partitions), with rules such as "parts must differ by at least two". The other side multiplies together a neat pattern of terms. That two such different calculations always give the same answer is the kind of thing mathematicians find beautiful — and mysterious.
A knot hiding in the equations
Now a team led by Yifeng Huang, a postdoctoral researcher at Emory University in the United States, working with Kenny Lau and Ken Ono, has shown why the identities are true. Their proof, posted to the preprint server arXiv in September 2026 (arXiv:2609.20567), connects the identities to the geometry of curves described by the equation x to the power a equals y to the power b, where a and b are whole numbers with no common factor.
Here is the surprising part. If you take one of these curves and look at where it pierces a tiny sphere around its central point, the curve traces out a knot — a loop winding around the sphere. For the classic Rogers–Ramanujan case, that knot is the trefoil, the simple three-lobed knot you see in Celtic designs. The identities, it turns out, are really statements about the geometry of that knot.
And the trefoil is only the first knot in the family. For every pair of numbers a and b, there is a more intricately wound knot — a torus knot — and each one gives a new identity of the same kind. The case a = 2 was already known: it contains the Rogers–Ramanujan identities and the Andrews–Gordon identities found in the 1960s. For a = 3 and beyond, nothing was known. The new work proves the whole pattern, so there are now infinitely many families of these identities, one for every value of a.
A letter-writing campaign
The story behind the proof is almost as good as the maths. Huang began the work while a graduate student at the University of Michigan, writing letters — more than 40 of them — to mathematicians around the world, trying to persuade someone that his idea was worth pursuing. One recipient was Alexei Oblomkov at the University of Massachusetts Amherst, whose work connects geometry and knot theory. Together with Ruofan Jiang, now at the University of California, Berkeley, Huang and Oblomkov formed the conjecture that the new proof now settles.
Ken Ono, one of the world's leading experts on Ramanujan, was another letter recipient. He first found Huang's ideas hard to accept, then joined the work. The team also used an AI-based mathematics system over several months to help develop and check the proof, and the final identities have been formally checked in the Lean proof system.
For students, the lesson is a hopeful one: a famous result can hide a completely different explanation for a century, and a young mathematician with a bold idea — and the persistence to write 40 letters about it — can be the one to find it.
If your child loves puzzles, patterns and the "why is that true?" behind a formula, a first lesson in maths can start with partitions and knots and end with how mathematicians prove things that look impossible. Book your first lesson
Sources:
- Huang, Y., Lau, K. and Ono, K., "Rogers–Ramanujan identities from the geometry of X^a=Y^b", arXiv:2609.20567, September 2026: https://arxiv.org/abs/2609.20567
- Lau, K. and Ono, K., "Modularity of Point Counts for the Curves X^a=Y^b: New Rogers–Ramanujan Identities", arXiv:2608.05480, August 2026: https://arxiv.org/abs/2608.05480
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