At the end of the 19th century, soon after Hamilton’s death in 1865, William Thomson (who would eventually become Lord Kelvin) had a spark of inspiration. He had the magnificent idea that chemical elements could be classified through their internal structure. Based on the earlier work of Hermann von Helmholtz in fluid mechanics and the investigations of his friend, the Scottish natural philosopher Peter Tait, on smoke rings, Thomson believed that atoms are composed of tangled vortices. In Thomson’s vortex‐atom theory, atoms are knots of increasing complexity that can be deformed but can never lose their intrinsic shape: if a string is knotted and closed on itself, it can be deformed but its knottiness cannot be changed without cutting it. To each element there would be a corresponding knot. The problem was that, at the time, there was no classification of knots despite some early work by Carl Friedrich Gauss and Johann Benedict Listing on the topic.

At Thomson’s suggestion, Tait and Little developed tools to study knots and they undertook a systematic enumeration of knots organized by their crossing numbers. Mathematically, knots are closed curves in space with no self‐intersection (technically, a knot is an embedding of a circle in three‐dimensional Euclidean space). A mathematical knot is very similar to the idea of a knot obtained by knotting a string and closing it. Two knots are the same if one can transform one into another by deforming and stretching the string but without letting the string go through itself or cutting it. Tait organized knots by their crossing numbers, the smallest possible number of intersections found in a regular projection (that is, by looking at all the possible shadows of the knot on a plane and disregarding the irregular shadows where three or more curves meet at a crossing point). The simplest knot, the unknot shown has zero crossings. The trefoil knot 31 has three crossings, and so on. Tait’s main classification is concerned with prime knots, which are simpler than the so‐called composite knots. Composite knots are obtained as the sum of prime knots, such as the granny knots and the square knots shown in Figure 33. Prime knots can be used as the basic building blocks of knots, just as prime numbers are used to decompose any positive integer.

The square knot can be obtained as the sum of a trefoil knot and its mirror image. The granny knot is the sum of two identical trefoil knots. Composite knots are obtained by cutting prime knots and gluing together their segments, as shown for both knots.

A table of prime knots of up to six crossings. The first number defines the minimal number of crossing and the second number is used for classification: 62 designates the second knot with 6 crossings in the table of knots. The popular name for these knots are: the unknot 01, the trefoil 31, the figure-eight 41, the cinquefoil 51, the three-twist 52, and the stevedore 61.

Unfortunately for Tait and Little, by the time they completed their tables of knots up to ten crossings, the Russian chemist Dmitri Mendeleev had solved the classification problem with his periodic table of elements. The vortex‐atom theory was relegated to the dustbin of history. But their hard work was not in vain because a new mathematical theory was born and, in the 20th century, the study of knots became a main focus of research in low‐dimensional topology with surprising connections to other mathematical theories such as algebra and analysis. The topic is so relevant that one of the 1990 Fields Medals was awarded to Vaughan Jones for his contribution to knot theory and, in particular, for the invention of a function for each knot type, the Jones polynomial. In essence, a Jones polynomial P(x) is defined for each knot (for instance, for the unknot P = 1 and for the trefoil P = x + x3 − x4). Operations on knots can be translated into operations on polynomials, henceforth linking geometry to algebra. This beautiful construction is intimately related to other mathematical theories and allows for the distinction between two knots (if two knots have different Jones polynomials, they are necessarily different). Thanks to such constructions, it is now possible to give precise meanings to surgical operations on knots such as inverting a crossing by cutting a strand, letting another strand pass through and reconnecting the cut strand afterwards, as shown in Figure 34. This operation may change the topology of the knot by changing its knot type.

Surgery on knots. The knot type can be changed by swapping the crossing (from an over‐crossing to an under‐crossing). In three dimensions, this operation can be performed by cutting, passing the strand, and re‐tying the curve. After swapping a crossing, the knot has changed knot type from a trefoil to the unknot.

Nowadays, knot theory is still very much at the forefront of pure mathematics (see Figure 35 for a challenging problem). Yet, since the 1980s its results have found unexpected applications in the study of many physical and biological systems. Following the seminal discovery of DNA structure by Crick and Watson, as discussed in Chapter 6, it is now understood that DNA molecules carry the fundamental genetic information used by all living organisms for growth, development, proper functioning, and reproduction. Physically, the cellular DNA found in humans is an extremely long and thin chain. If straightened out, the largest human chromosome, chromosome number 1, would be 85 millimetres long and about 2 nanometres wide, a chain that is 42.5 million times longer than its diameter. These chains are tightly packed inside the nucleus and must be manipulated for key processes such as recombination, replication, and transcription to take place. DNA needs to solve not only a geometric packing problem but also many topological problems.

The unknotting problem consists in proving that a knot (left) is actually the unknot (right). Here is one of Wolfgang Haken’s Gordian unknots.

DNA is composed of two complementary twisted strands. During cell division, the strands must be completely separated. To do so requires completely unzipping the DNA and changing its knot type. Imagine that two twisted strings are attached at one end and you need to untwist them. If you just pull them apart, the twist in the two chains must be removed in order for them to be separated or the chain becomes overwound at the separation fork which prevents you from further separation. The solution to this geometric (and physical) impasse is to reduce the twist in the two strands of the DNA. This is accomplished in the cell by a special group of enzymes called type I topoisomerases, first discovered by James C. Wang in the 1970s (the prefix ‘topo’ stands for the topological action of the enzymes on the DNA, the ultimate mathematical enzyme!). Exactly like the surgical operation shown in Figure 34, these proteins cut one strand of a DNA double helix, let the other strand pass through, and then the cut strand is retied, reducing the overall twist

Topoisomerase 1 action on DNA. One of the braids of DNA is cut, passed through, and re‐tied. The overall action relaxes the twist of the two strands allowing for DNA replication.

Another important application of knot theory is in understanding the action of enzymes on DNA. For instance, recombinases are enzymes that manipulate the structure of genomes to control gene expression. These enzymes are known to break the DNA apart at specific recombination sites and to recombine the ends by exchanging them. There is no direct observational method for enzyme action but recombinases are known to change the topology of DNA. Therefore, one can synthesize circular duplex DNA and study the enzyme mechanism by its action on DNA topology (see Figure 37). By looking theoretically and experimentally at the products created by recombination, mathematicians working with biologists are able to identify the mathematical operations that mirror the action of enzymes, hence uncovering key mechanisms taking place in the cell.

Action of the resolvase Tn3. The recombination sites are the two arrows bound by the resolvase (grey circle). During recombination, the resolvase transforms a circular duplex DNA into a Hopf link. Successive recombinations lead to the figure‐eight knot and the Whitehead link. The next iteration would lead to the 62 knot (not shown).

The use of knot theory to study key processes in cellular biology was a pleasant surprise for many scientists. It also put new vigour into the theory of knots and transformed a rather esoteric subject into a mainstream applied topic. Nicholas Cozzarelli, a molecular biologist at the University of California Berkeley and one of the main proponents of topological enzymology once said:

Before Jones, the math was incredibly arcane. The way the knots were classified had nothing to do with biology, but now you can calculate the things important to you.