Knot theory is a theoretical study in mathematics, where your brain thinks of an imaginary knot, and manipulates it to your heart’s desire. Yes, the kind of knot you are probably thinking of now, it might be a shoelace, a knot in a piece of string or some utility knot. Good job, but it’s missing one detail: the knot needs to be tied at its ends. Think of this as a string with both ends tied together so that it can’t come undone when you play with it. Now you can pull at and twist this knot, as long as you don’t break it. Congratulations, you now understand the basics of knot theory. (1)

So why should we care about a niche field of maths that you will probably never use in your everyday life? Well, the first answer to that is simply ‘for the love of the game’. For some people problem-solving is an endless endeavour that satisfies an urge to understand and be intellectually stimulated. But that’s not for everyone. So then we remember all the times when random elements of pure mathematics became essential when applied to seemingly unrelated topics.

Such as how number theory became applied to information transmission, cryptography and computing. (2)

How quaternions made for more efficient digital transformations in computer science. (3)

Or how graph theory was used to strongly conjecture that any two people have 6 degrees of separation between each other. (4)

Although we may not routinely ponder these discoveries, it is because of the works of pure mathematicians that we can admire certain facts that we could not prove otherwise or appreciate how they silently helped to make all the digital devices in your homes.

But before we get into the applications, it is good to be familiar with some general terminology. That knot which you pictured earlier with its ends tied is called a standard knot. In 1867 Lord Kelvin thought of the revolutionary idea that what we know as elements - the ones made of protons and neutrons - are actually types of standard knots. (5) He wasn’t right, but it inspired his assistant Peter Guthrie Tait to begin the rigorous study of knots and we have been trying to find applications ever since. Here are the first knots in the greater sequence of the periodic table of knots (see cover image for more!):

Figure 1. An ordered table of the first 15 prime knots. (6)

There are knots made from one piece of string (prime knots) and knots made from multiple knots joined end-to-end (composite knots) (Fig.2b). There are also links, where two closed knots are combined without gluing the string (Fig.2a). Understanding any further implications of this terminology is not necessary here, but it may help to have a visual understanding of them for the next part.

Figure 2. a) Showcasing types of mathematical links; unlink on the left, Hopf link in the centre and whitehead link on the left. b) Demonstrating how two prime knots are combined into a composite knot. c) Demonstrating chirality in trefoil knots, notice the overlapping pattern.

Lastly, like many things in mathematics we need a way to systematically and efficiently describe how we manipulate the knots. Luckily, Kurt Reidemeister had the pleasure of providing us with a knot-manipulating moveset in the 1930s through rigorous proofs.These are the legal set of moves that can be done to a knot without changing the knot structure. If we were to cut the knot, twist or untwist the string and then reattach the ends, this is called a crossing switch and it changes the knot. Again, this is not an extensive course but it helps to know of the terminology and visualise it. Feel free to do more research into the details of these topics using the references below!

Figure 3. A depiction of the Reidemeister moves.

DNA and knot theory

Deoxyribonucleic acid (DNA) is the most important and relevant knotting molecule. Each cell nucleus contains (on the millionth order) DNA that is regularly knotting, coiling and compressing to fit into this tight space. However, the best application of knot theory is to the closed end, circular DNA in bacteria. During DNA replication, the unwinding of DNA at one end creates immense torsional strain on the other side of the loop, which is enough supercoiling that prevents replication and leads to cell death.To counter this, bacteria utilise an enzyme known as type II topoisomerase which makes double-stranded cuts in the DNA, followed by a rearrangement of the tangle and reconnecting of the strands, a crossing switch! Without this adaptation, all cellular life would have evolved differently.

If you gave this DNA to a mathematician and asked which position in the DNA would be best for this enzyme to cut with the intent of untangling, they could spend a lifetime performing Reidemeister moves and contemplating, never knowing where or how many cuts to make. In contrast to our world’s best mathematicians, topoisomerase is incredibly efficient in where it cuts. We have yet to understand what mechanism allows for such accurate cuts, but practical research into topoisomerase could potentially help knot theorists solve the immensely inscrutable question of the minimum number of crossing switches to simplify any knot. Furthermore, if an understanding of the mechanisms for topoisomerases in bacteria and humans is possible, then humanity can access a new form of control over DNA. It has been speculated that there are possible uses of topoisomerases to inhibit cancer growth, or as a revolutionary way to treat bacterial disease. While we do not have this intel right now, this is one of the ways knot theory could be integral to applied sciences and given time and research funding, it can prove itself useful. (7-8)

Knots in chemistry

So what other molecules can form knots? Chemists have been creating molecules which involve the basic knots and links since the 1960s (see Fig 4), when topological isomerism was discovered and characterised. Topological isomers are chemicals that are similar in many properties, but differ in spatial arrangement. We can think of it like chirality for knots (see Fig 2c). Chirality is the property of an object not being the same as its mirror image, like a right and left hand. Subsequently, these molecules were made through a technique called ‘templating’, where a metal ion or some template structure was used to produce a desired product, based on how the template interacts with the reactants. There is also another category of knot called a ravel (Fig 4h), where a knot has multiple strings connected at vertices. Altogether, the study of topological isomerism and templating techniques have been advanced by the experimental desire to produce these beautiful molecules. This then indirectly contributes to the production of new molecules and drugs that can go on to have real world impacts. (9)

Figure 4. a) The first molecular trefoil knot produced in 1989. c) The first molecule pentafoil knot produced in 2011. d) First molecular Borromean rings, a type of link produced in 2004. f) The first molecule solomon link produced in 2013. h) The first molecular ravel produced in 2011. (9)

The recent breakthrough in knot theory

I admit, progress in knot theory is slow and perhaps you did not find the scientific revelation of knot theory here that you were hoping for. But that does not mean that current research is ineffective. As recent as June of this year, there was a groundbreaking proof. Think back to the prime and composite knots (scroll up if you have to). Prime knots have an unknotting number, which is the number of crossing changes needed to simplify it to the unknot, similar to what the topoisomerase does. If we merge two prime knots into a composite knot, it can be easily seen that it takes as many crossing switches to simplify the composite, as it does the crossing switches for the sum of the primes. In other words, to untangle a composite knot, you cut and reglue it as many times as the prime knots that make it up. Now, the breakthrough was a proof that it is possible to untangle some composite knots through less crossing switches than the sum of its prime knots. This may seem bleak, but it disproves a widely believed conjecture and now theorists are one step closer to solving the question of the minimum number of crossing switches needed to simplify a knot. (10) 

Conclusion

I will end this with a quote from Dr Arumina Ray, a mathematician that specialises in knot theory and low-dimensional topology at the University of Melbourne, and a dear professor of mine. Hopefully this is just more proof (pun intended) that the work us mathematicians do is tangible:

“I had never imagined that mathematics could be used to describe something so abstract as knot theory, but to me the appeal was its tangibility. No matter who you are, there really is something in mathematics for you.”

References

1. Pencovitch M. What’s not to love? [Internet] Mathematics Today. 2021. Available from: https://ima.org.uk/17434/whats-knot-to-love/ 

2. Koblitz N. A course in number theory and cryptography. 2nd ed. Springer Science & Business Media; 1994. 

3. Jeremiah. Understanding quaternions. 3D Game Engine Programming [Internet]. June 25, 2012. Available from: https://www.3dgep.com/understanding-quaternions/ 

4. Zhang L, Tu W. Six degrees of separation in online society [Internet]. Research Gate. 2009. Available from: https://www.researchgate.net/publication/255614427_Six_Degrees_of_Separation_in_Online_Society 

5. Wilson RM. Holograms tie optical vortices in knots. Physics Today. 2010. https://doi.org/10.1063/1.3366639

6. Li M, Wang T, Kau A, George W, Petrenko A. Knots. Brilliant. 2025 [Internet]. Available from: https://brilliant.org/wiki/knots/ 

7. Catherine. All tangled up: an introduction to knot theory [Internet]. Gleammath. April 28, 2021. Available from: https://www.gleammath.com/post/all-tangled-up-an-introduction-to-knot-theory 

8. Skjeltorp AT, Clausen S, Helgesen G, Pieranski P. Knots and applications to biology, chemistry and physics. In: Riste T, Sherrington D, editors. Physics of Biomaterials: Fluctuations, Selfassembly and Evolution. Dordrecht: Springer Netherlands; 1996. p.187–217. https://doi.org/10.1007/978-94-009-1722-4_8

9. Horner KE, Miller MA, Steed JW, Sutcliffe PM. Knot theory in modern chemistry [Internet]. Chemical Society Reviews. 2016;45(23). Available from: https://durham-repository.worktribe.com/output/1394834

10. Brittenham M, Hermiller S. Unknotting number is not additive under connected sum [Internet]. Arxiv. 2025. Available from: https://arxiv.org/html/2506.24088v1