The Topological Theory of Knots is the object of an active mathematical investigation that, like the Gordian knot, still awaits the Alexandrian sword capable of delivering the final blow.
What is a knot?
We have all had the experience of unraveling a piece of string or an electrical extension cord that has not been used for a long time. After completing such a difficult task, we found that, in reality, there were few knots and that most of the mess was resolved by stretching both ends of the rope, which leads us to distinguish a knotted rope from another that is simply tangled.
Suppose that we have a piece of rope and that we make the knot as simple as possible and that, once this is done, we take both ends of the rope and join them, but not with a new knot, because this would be the story of never-ending, but sticking them so that there is no way to know where the union is. This is the way to get a knot in the mathematical sense of the term. 
Knots that have managed to apply some kind of mathematical technique to classify them are often colloquially called domesticated, the others, which are many, are called savages. The knot can be as complicated as you like. The number of convoluted crosses we can make or how long the rope is does not matter when defining the knot. If we now carefully place it on a table we will have carried out, although it is hard to believe, a sophisticated mathematical operation that receives the pompous name of “regular projection of the knot on a plane”. If we want to have a graphical representation, what is called a “knot diagram”, it is enough to keep in mind that at the point where a piece of the string passes under another, a point called a “crossing” or crossing,
If we look at the previous diagram we see that to go from the first figure to the second one it is enough to make a bend, that is to say, that we do not need to cut the rope and reattach it, nor those kinds of things that are so hateful in the eyes of a topologist. Two knots are said to be “equivalent” if and only if they can be passed from one to the other by continuous deformation, bending, stretching, etc., but never cutting. For example, between the next two knots, there is no continuous transformation that allows us to move from one to the other.
One of the herculean tasks proposed by knot theory is to classify them all, for which the first thing to decide is when two knots are the same. But since in Mathematics a thing can only be equal to itself, a broader criterion is established, that of equivalence.
The rigorous definition of equivalent knots is somewhat complex and requires more sophisticated topological concepts than those presented here, so we will stick with the intuitive idea of continuous transformation that we have discussed before. Accordingly, the two figures in the first diagram are equivalent knots. In reality, there is no knot, which is why the circle or any knot equivalent to it is called a “trivial knot”. Although it is possible to acquire some visual ability to distinguish equivalent knots, it is obvious that the classification of knots is not something that can be done by eye. It is not easy, for example, to see that all these knots are different.
Or that these two are equivalent:
Is there a knot?
Before undertaking the enormous task of classifying all the existing knots, mathematicians, who are capable of doubting anything, first ask themselves if there is any knot, that is, a knot that is not equivalent to a circle. This question was solved by the German mathematician Kurt F. Reidemeister (1893-1971) in a very ingenious way. The first thing he did was establish three basic operations, three movements, using which a knot can be made or undone, which are twisting, overlapping and sliding, and, naturally, the inverse movements of these three.
From here, Reidemeister opened the color box and took three to paint the knots, strictly following the following two rules:
- Only two different colors can appear in any intersection.
- To color a knot you have to use at least two colors.
When a knot can be painted with three colors following these rules, it is said, making a slight abuse of language, which can be tricolor. For example, the so-called clover knot can be tricolored, something that cannot be done with the trivial knot.
The grace of the method is that the property of being tricoloreable is invariant by the transformations of torsion, superposition, and slip described above. That is, it is a topological invariant that allows us to affirm that all three-color knots are topologically equivalent to each other, as is the case, for example, with the following three knots.
We are now in a position to state that there is at least one knotting different from the trivial.
The Reidemeister method is extensible to a larger number of colors and, more interestingly, it can match an integer (corresponding to the number of colors) to a given class of knots. It is not a definitive method to classify all possible knots as it has some limitations. You cannot, for example, distinguish a clover on the left from one on the right, as they are both tricolor and yet are different knots.
The first method of classifying knots is to calculate the “order” of the knot, which is the number of times the rope crosses itself. Through this system, it has become known that there is only one node with 3 crosses, 2 with 5, 3 with 6, 7 with 7, 21 with 8, 49 with 9, and 165 with 10. In 1998 it was determined, with the help of powerful computers, which exist a total of 1,701,936 knots with 16 crossings or less.
It is understood that when defining the order of a knot we refer to the minimum number of crosses that the knot has since the rope could be entangled with loops that were not authentic knots. The first job then is to untangle the rope, something that can be done using the three basic Reidemeister movements described above. And also, thanks to a theorem from mathematicians Joel Hass and Jeffrey C. Lagarias, we can already know what is the minimum number of steps necessary to untangle a string of N crosses:
That is, we are guaranteed that we can untangle a rope with N crosses in less than 2 raised to one hundred billion per N. An operation that may take us longer than the age of the Universe, but in a finite number of steps, which is what counts for the mathematician.
Being able to find a criterion to classify knots is the same as looking for invariants, that is, finding some property that remains unchanged when we properly transform the knot. Among the most important invariants are polynomials. The first appeared in 1928 and were introduced by the American mathematician James W. Alexander (1888-1971). It is a simple invariant that associates a polynomial with a certain class of equivalent knots. For example, the following knot has an Alexander polynomial x 2 – 3x +1.
Instead, this other, called the Savoy knot, has the polynomial x 2 – x +1.
Even though it is an efficient method to characterize equivalent knots, it has the serious drawback that there are different pairs of knots with the same Alexander polynomial.
Over time, other invariant polynomials appeared, such as those introduced by JH Conway (1937-) in the early 1960s with the help of computers, or those established in 1984 by the New Zealand mathematician Vaughan Jones (1952-), capable of distinguishing enter the cloverleaf knot to the left or right. These polynomials are respectively characterized by x + x 3 – x 4 and x -1 + x -3 – x -4 .
There are thus several mathematical criteria for the classification of knots, but none of them is complete, in the sense that it achieves a general classification of all possible knots. The issue of classification of topological nodes, therefore, remains an open problem.
What are knots for?
Jones was the first to note that there was a close relationship between statistical mechanics, a branch of physics that studies the nature of gases and liquids as large sets of atoms, and the polynomials associated with knots. Later, Louis H. Kaufmann found an interpretation of the Jones polynomial in terms of a state function, something that is limited to statistical mechanics, thus inaugurating a new strand in knot theory that is called the Combinatorial Theory of Knots. This application of Physics to Mathematics has been relevant in the sense that it inverts the historical process in which Mathematics always provides Physics with its logical framework.
Currently, knot theory finds applications in areas as diverse as electrical circuit analysis or cryptography. It has also been shown to be very useful in modeling the physics of polymers and liquid crystals and, in general, in all those situations in which knots appear between networks or meshes.
And in another order of things, although also limited to the field of physics, there are currently great expectations that the Theory of Knots provides the Physical Theory of Strings with the necessary complement to give a unified description of the four fundamental forces of nature: gravity, electromagnetism and the strong and weak interactions between particles.
The curious fact is that in the late nineteenth century, William Thomson, better known as Lord Kelvin, built a theory that matter was made up of vortices that bonded and knotted together in a fluid medium called ether. A century later the theory is rescued on the stage of quantum mechanics, in which knots and strings are also the origins of matter.
But perhaps the most important application of Knot Theory to other sciences is that which takes place in the domain of molecular biology. Something that basically should not surprise us, since if we think about the possibility of locating something that is one meter long in a space of about five-millionths of a meter, we are going to have to roll, tighten and crisscross this object so that it can fit in such a small space. This is what happens with a molecule of human DNA. It is therefore not surprising that knots appear in the double helix structures of the genetic material and that the node topology has become an essential tool in this area of research.
It is also worth mentioning, even as a curiosity, an application of Theory of Knots that, despite its overwhelming number of detractors, still have unconditional followers. In the mid-1970s, Jacques-Marie Émile Lacan (1901-1981), a French psychiatrist and psychoanalyst, was subdued after a conversation with a young French mathematician (Valérie Marchand) by the so-called Rings of Borromeo (explained below). From that moment on, Lacan established intense contacts with other mathematicians who introduced him to knot theory, thus initiating a long adventure that would lead to the conjunction between Topology and Psychoanalysis. Currently, there are still seminars to introduce psychoanalysts to the secrets of topological knots,
Borromeo’s Rings are three rings intertwined in such a way that no pair of them is linked. But it is impossible to separate them.
It is an interlacing of three trivial knots (three circumferences or, more generally, three closed curves without knotting). It is a symbol that can be found in some companies as a logo. Since it represents the cohesion force of a group. Only if one of the rings is cut, the entire figure is unlinked. Its name dates back to a family of Italian Renaissance princes who adopted it as a coat of arms.