Forget Me Knot: A Mathematician’s Lasting Tryst with Knot Theory

Forty years since the discovery of the Jones Polynomial, Shantha Bhushan reflects on her enduring relationship with knots and braids.

Shantha and her student origami min

Buried in an obscure nook of Youtube is an old recording of mathematician Joan Birman. In it, the American topologist looks back on a particularly eventful week in her life, which began with a visit from a young Kiwi researcher Vaughan Jones, and ended with the discovery of the Jones polynomial”. This was 40 years ago in 1984.

We both understood that it was bound to be a major discovery in knot theory,” wrote the now 96-year-old Birman, in a tribute to Jones, who, unfortunately, passed away in 2020. According to another obituary of Vaughan Jones that appeared on Nature, the polynomial was the kind of tool that topologists had been seeking for decades.” A tool that would enable them to distinguish different types of knots. 

Bitten by the knot bug

Knot theory is a sub-field of topology, which is the mathematical study of objects that can be twisted, stretched, or deformed in some way. When Shantha Bhushan started her PhD in topology at IIT Bombay, the maths world was still abuzz with the discovery of the Jones’ polynomial. Knot theory was an exciting field to be in, in the late eighties. Though Shantha’s dissertation focused on a different area of topology, she too was bitten by the knot bug. 

When we think of knots, we think of shoelaces, gift-wrapping, sailors… or we think of crochet yarn, or maybe surgeons tying up a knot after stitching up something,” says Shantha, who now teaches mathematics at Azim Premji University. These are all examples of open” knots, meaning that the two ends of the rope/​string/​ribbon/​lace are separate. If not for physical factors such as friction and tension, open knots would easily come undone. Topologists like Shantha are more interested in closed” knots, which have no free ends. 

An open knot (above) has free ends, unlike a closed knot (below). It’s the latter that is of most interest to mathematicians. 

Credit: the​-gist​.org

She illustrates what this would look like: Imagine knotting up a metal wire and welding their ends together. Now there is less wiggle room, and things get interesting!” 

Shantha then coaxes us to do a similar exercise with another piece of wire, and then try to figure out if the knots are identical. You can stretch the wires, go over, under, basically perform any operation, as long as you don’t cut them. If the two knots are equal you will be able to transform one into the other,” she states. This is a fundamental aspect of knot theory.

The first documented mathematician to study knots was — incidentally, Shantha’s favourite — Carl Friedrich Gauss in 1794. But the quest to mathematically distinguish them seems to have started nearly a century later. This was when mathematicians Lord Kelvin and Guthrie Tait together came up with the idea that atoms and the universe were made out of a material called ether, and the differing properties of atoms (for example hydrogen and helium) were due to the distinct ways ether knotted up.

Though the existence of ether was soon disproved by the famous Michelson – Morley experiment in 1887, this did not end Tait’s fascination with knots. He went on to create a categorisation system for knots. And thus, out of an attempt to understand what this universe is made of, knot theory was born. 

Tait’s classification of knots up to 7 crossings. Credit: Wikimedia Commons

Tait’s classification of knots was based on the minimum number of crossings a knot involved. The simpler a knot, the fewer times its lines crossed each other in a 2‑D diagram (as seen above). He came up with a table that depicted how many different knots could be formed with n’ number of crossings. 

For example, the simplest knot, called an unknot’, is basically a circular loop with 0 crossings; there are no knots with 1 or 2 crossings; there is one knot each possible with 3 and 4 crossings respectively; 2 with 5 crossings; 3 with 6 crossings; and 7 with 7 crossings. That’s as far as Tait got, which itself was pretty impressive considering there were no theoretical techniques at his disposal, only his keen intuition.

Later, with the help of Thomas Kirkman and Charles Little, Tait’s table went up to 10 crossings — there were 165 different knots with 10 crossings. In 2020, Australian mathematician Benjamin Burton classified all (prime) knots up to 19 crossings. How many types of knots exist with 19 crossings? Close to 300 million! 

Once you get a sense of the number of possible knots, it becomes easier to appreciate the value of Joan Birman and Vaughan Jones’s contribution. Two knots may have the same crossing number, but are they identical? That was extremely hard to say, but the development of the Jones polynomial could do exactly this. The discovery was huge. I was very excited,” recalls Shantha. 

Coral reefs, crochet and curvature

Though Shantha was not able to formally pursue the subject during her PhD, knots and braids continued to occupy the mathematician’s mind. Her crocheting and knitting hobby deepened her interest. Though both crocheting and knitting involve open knots, they do offer a lot of opportunities to grasp complex concepts in maths, particularly in topology.

The connection [between mathematics and crochet] came to me when I read about a professor from Cornell University who used crochet to show how to create surfaces with negative curvature — meaning they bend inwards like coral reefs,” she says. As a PhD student in topology, Shantha had been introduced to the concept of negative curvature, but only in the form of mathematical models and functions. When I first used crochet to create such a surface, I finally felt like I understood what was going on! It felt very empowering.” she reminisces. 

When I first used crochet to create a negative curvature surface, I finally felt like I understood what was going on! It felt very empowering. Now, as a teacher, my job is to help students not fear maths. I see anything tactile and visual, as a way of freeing students from this anxiety and fear.”

Examples of surfaces with negative curvature: (left) corals under the ocean and (right) a crocheted pattern. 

Credit: University of Alabama at Birmingham and Flickr/​eraphernalia_​vintage

This experience created an indelible mark on Shantha, both personally and professionally. A PhD is a very difficult time for everybody, and it is worse when you can’t visualise something… you start to believe you are not good enough.” A similar feeling haunts many undergraduate students of mathematics as well, she noted as her career transitioned from research to teaching. Now, as a teacher, my job is to help students not fear maths. I see anything tactile and visual, as a way of freeing students from this anxiety and fear.”

One powerful tool Shantha and her colleagues in the University use with students is origami. There is a lot of deep maths that origami can introduce us to. It’s not that doing origami can make you understand everything, but at least it removes some barriers,” she insists. And to a significant degree, Shantha was vindicated when their teaching experiments began to show results.

We can see a shift in the way students are thinking about maths. It’s no longer just a bunch of formulae and symbols that only very smart people can deal with. It’s something that they can play with, enjoy,” says Shantha. It was particularly moving for the maths group to witness this shift taking place among students from disadvantaged backgrounds. Many of them take maths because they are told that they can get a job in software. To see their relationship with maths improving is very rewarding.”

One powerful tool Shantha and her colleagues in the University use with students is origami. There is a lot of deep maths that origami can introduce us to. It’s not that doing origami can make you understand everything, but at least it removes some barriers,” she insists.

Students learn maths from origami during Shantha and Divakarans classes. The Mathematics faculty incorporates many such diverse activities to make Maths learning fun.

Craft as a gateway to higher mathematics

The potential of arts and crafts as a gateway to higher mathematics is by now well known, even if understudied. Applied mathematician Elisabetta Matsumoto has been quoted as saying knit theory is knot theory”. The American Mathematical Society includes special sessions on maths education and fibre arts quite regularly.

Looking back at the 40 years since the discovery of the Jones polynomial, it’s clear that the hype was not for nothing. Shantha compares it to the 4‑minute mile, first recorded by a British athlete in 1954. After Roger Bannister proved it was achievable, many others were able to cross the four-minute barrier. Similarly, the Jones polynomial opened the floodgates to several other types of knot-distinguishing polynomials such as HOMFLYPT and the Khovanov invariant.” 

Today, knot theory is no longer merely a cool theoretical field. It is being applied in cryptography, molecular biology and synthetic chemistry. Shantha added: In the last few decades, quantum computing and knot invariants has led to quantum topology. It is amazing to see the interplay and how growth in one field influences the growth in the other!” 

About Shantha

Shantha Bhushan is a faculty member at Azim Premji University. She holds a PhD from IIT Bombay and was a DST fellow at IISER Pune. She has also worked on knot theory in understanding protein backbones. 

About the Author

Nandita Jayaraj is a Science writer and Communications Consultant at Azim Premji University.

Know more about the BSc in Mathematics programme at Azim Premji University.