Ramanujan equations: mathematicians link his identities to knots, find infinitely many
By Devansh TiwariKnowledge Systems Writer, Peepals Sanskriti
Representative image: geralt / Pixabay
New work by Yifeng Huang, Ken Ono and colleagues, reported on 30 October 2026, shows the Rogers-Ramanujan identities are one case of an infinite family tied to knots.
What Happened
Mathematicians have shown that two famous Ramanujan equations, the Rogers-Ramanujan identities, are a single case of a much larger, infinite family of similar formulas, and that each member of the family is linked to a knot. The work, led by Yifeng Huang, a postdoctoral fellow at Emory University in the United States, was described in a report published on 30 October 2026. It rests on a proof that Huang, Kenny Lau and Ken Ono of the University of Virginia posted as a preprint on 17 September 2026.
Key Facts
- The Rogers-Ramanujan identities are two formulas that appeared in the letters of the Indian mathematician Srinivasa Ramanujan; they are named after him and the mathematician Leonard James Rogers.
- Huang's idea was that the two identities correspond to a trefoil knot, the simplest true knot, in a particular mathematical space.
- His collaborations produced the Huang-Jiang-Oblomkov (HJO) conjecture: for every curve of the form x^a = y^b, a count of certain matrix solutions always collapses into a Rogers-Ramanujan-type identity.
- The September 2026 preprint by Huang, Lau and Ono, titled "Rogers-Ramanujan identities from the geometry of X^a=Y^b", proves that conjecture for every such curve.
- An earlier preprint, posted in August 2026 with Peter Paule, set out an algebraic-geometry framework for the identities.
- The proof comes with a computer-checked version written in Lean, a program that verifies each logical step; one review notes that this check relies on two results taken from earlier literature.
Why It Matters
Srinivasa Ramanujan is one of India's best-known mathematicians, and his notebooks and letters are still a source of new research. Many of his results were written down with little or no proof, and mathematicians have spent decades working out why they are true. The Rogers-Ramanujan identities are among the most famous of these. Each one says that an infinite sum and an infinite product, built in very different ways, are exactly equal. Such formulas matter in counting problems, for example in counting the ways a whole number can be split into smaller parts, known as partitions.
For a long time it has been known that more identities of this type exist; an infinite family was found by mathematician George Andrews in the early 1970s, according to the report. What is new is the source the researchers point to. Huang's approach treats each identity as a count of solutions to a simple equation of the form x^a = y^b, where the solutions are matrices (grids of numbers) rather than ordinary numbers. Each such curve is related to a torus knot, a knot that can be drawn on the surface of a doughnut shape. The trefoil gives back Ramanujan's original pair, and other knots give new identities, so the family never runs out.
The report describes how Ken Ono, a specialist in Ramanujan's work, at first dismissed Huang's idea as "crazy" when he received a letter about it, then later called it "mind-blowing". That change of mind captures how unexpected the link between Ramanujan's formulas and the shape of knots seemed to experts.
For Sanskriti readers interested in India's knowledge traditions, the result is a reminder that Ramanujan's work is not only history. Formulas he wrote down more than a century ago continue to open new questions in geometry and number theory.
| Paper | Posted | What it does |
|---|---|---|
| Algebraic geometric framework of Rogers-Ramanujan identities | August 2026 | Sets out the geometry behind the identities |
| Rogers-Ramanujan identities from the geometry of X^a=Y^b | 17 September 2026 | Proves the HJO conjecture for every such curve |
| Lean formalisation | With the September preprint | Computer check of the proof, relying on two earlier published results |
Impact
Short-term: The papers are preprints, meaning they have been shared publicly but not yet through formal peer review in a journal. Other mathematicians will now test the arguments.
Long-term: If the proof holds, it gives a single geometric explanation for a large class of Ramanujan-type formulas and may help researchers find and prove new ones in a systematic way rather than one at a time.
Who is affected: Mathematicians working in number theory, combinatorics and algebraic geometry; students and teachers of mathematics; and readers who follow Ramanujan's legacy in India and abroad.
Key Takeaway
A September 2026 proof by Huang, Lau and Ono shows Ramanujan's two famous identities are the first member of an infinite family of formulas, each tied to a knot.
Questions and Answers
What are the Ramanujan equations that mathematicians found infinitely many of?
They are identities of the Rogers-Ramanujan type: formulas stating that a particular infinite sum equals a particular infinite product. Ramanujan's original pair corresponds to the trefoil knot, and the new work shows that other knots give infinitely many more formulas of the same kind.
Who did this research?
Yifeng Huang, a postdoctoral fellow at Emory University, developed the idea. The proof of the related Huang-Jiang-Oblomkov conjecture was posted on 17 September 2026 by Huang, Kenny Lau and Ken Ono of the University of Virginia.
Has the result been peer reviewed?
Not yet. The papers are preprints on the arXiv website. The proof does come with a computer-checked version in the Lean system, although that check relies on two results from earlier literature.
Why is Ramanujan's work still studied today?
Ramanujan wrote down many results without full proofs, and many of them later turned out to connect with deep areas of mathematics. Researchers continue to explain, extend and apply his formulas, as this work on knots shows.
Disclaimer: Prepared by the Peepals newsroom from publicly available sources with AI assistance. Information is accurate to the best of our knowledge at the time of publication and may change. Images may be representative. Not professional advice. Report an error via our contact page.
By Devansh Tiwari, Knowledge Systems WriterCovers Knowledge Systems and more for Peepals Sanskriti. All stories by Devansh Tiwari
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Devansh Tiwari is a pen name of the Peepals Global Editorial Team, which researches, fact-checks and publishes every story. The portrait is an illustration.
