How The Most Useless Branch of Math Could Save Your Life

- Most of us tie our shoelaces wrong. There are two ways to tie a knot in shoelaces. On one, go through the loop counterclockwise and on the other, go around the loop clockwise. These two methods seem al

most

identical, but one of these knots is far superior to the other. It doesn't loosen or come undone as easily. To understand why, we must delve deeper into knot theory. It is an entire

branch

of

math

ematics that aims to identify, categorize and understand all the possible knots that may exist. So far we have discovered the first 352,152,252 knots. Each one has its particular properties and characteristics.

I think it's fascinating that there is such a thing as a periodic table for knots, but it's not pure

math

ematics. Knot theory has proven remarkably useful. It is essential for the structure of proteins and DNA. It is giving rise to new materials that

could

be stronger than Kevlar. It is even used to develop medicines that save millions of lives. All this just to try to understand the humble knot. So what is a knot? Well, in our everyday lives we see knots like this one or this one, but if you try to study them rigorously, you want to be able to separate them so you can really see what's going on.

The problem is that knots like this are held together only by tension and friction. So if you pull too hard on them, they fall apart. So, to capture the knot of the rope, mathematicians came up with the idea of ​​connecting the two ends. And now, well, you can untie the knot to study it, but it will never fundamentally change. So in knot theory, all knots exist in closed loops. This means that the simplest knot you can have is simply a circle like this. Now, it is true that this is not a big knot, which is why it is called unknotting.

Here is another knot. Again, it is made from a single piece of rope that forms a closed loop. Here's another one. Two knots are only different if you can't turn one into the other without breaking the loop. This is the simplest knot after untying. It's called a cloverleaf, and you can see that I have no way to turn it back into a circle unless I break it, pull out the knot, and then close it again. Now I have two untied. It is surprisingly difficult to tell two knots apart with the naked eye. Here is a simple mystery knot.

Is it a knot, a clover, or neither? I'll give you a second to figure it out. It's actually a cloverleaf, which you can see if I untwist it and rearrange the knot a little bit. And our first tricky knot, well, actually is just untying. I'm going to try to untangle it so you can see it. There you go. It was just a loop. In fact, all of these are unleashed and this is where the problem begins. You can't just tangle a rope at random and connect the ends. To tie a new mathematical knot, you must show that it is not simply a tangled version of another knot.

So how do you tell the difference between two knots? This question, also known as the knot equivalence problem, is so difficult that it has driven the entire field of knot theory for more than 150 years. Alan Turing even wrote in his final publication: "There is still no known systematic method" by which one can tell whether two knots are the same. "A decision problem that

could

be unsolvable" is that of knots. "The results of this paper set certain limits "to what we can hope to achieve simply by reasoning." Previously, the

most

famous knot problem in history was the Gordian knot.

It was said that whoever untangled this enormous knotted rope was destined to It rules all of Asia, legend says, Alexander the Great just showed up and cut it off. That wouldn't be a valid solution in knot theory. And there are other famous knots in history that date back to the Indus Valley tablets. It was used in medieval Celtic designs, Chinese knots, and in Hinduism and Buddhism. In the Inca civilization, knots were tied into ropes called quipu to track everything from taxes to calendars of the House of Borromeo, a family. Italian noble that has been around since the 1300s.

Borromeon rings are technically a link, which is simply a knot with multiple loops of rope. The most basic link is the unlinking, two loops that. They are not actually connected, just like unknotting. After that is the Hopf link. Then, the Borromeon plays and more. But the problem of knot equivalence did not arise until centuries later. In January 1867, Scottish physicist Peter Guthrie Tait showed his homemade smoke machine to renowned scientist William Thomson, later Lord Kelvin. Tait had read a paper that said a vortex ring should be eternally stable in an ideal fluid. Intrigued, he set up two wooden boxes containing some kind of toxic mixture of ammonia, sulfuric acid and salt.

When he tapped a towel spread over the back of each box, chemical smoke came out of a circular cut in perfect rings. Kelvin, who was watching, was transfixed by the rings. He had been pondering the composition of atoms, a fundamental question of the time, and suddenly saw an answer. He declared that atoms must be made up of vortex rings of ether, a medium invisible everywhere. Different vortex ring nodes would form different elements. The shape of the Hopf bond explains the double spectral lines of sodium. The simple, knotless ring was hydrogen. Tait was skeptical, but when Kelvin's vortex model of the atom became a leading theory, Tait began investigating knots in earnest.

In his mind, he created a periodic table of the elements with each new knot he found. The crossing number is a simple way to classify nodes. Simply take the simplest form of knot, one with no extra twists or tangles, and count all

your

crossings. By hand, Tait discovered a three-cross knot, the shamrock, then a four-cross knot, the figure eight, then two five-cross knots, three six-cross knots, and seven seven-cross knots. A quick note: knots are additive. You can join several knots to make a new knot, such as combining these two clubs into a six-cross knot. This is called a compound knot, but some knots cannot be decomposed into simpler knots.

These are known as primary knots. Since all compound knots are simply constructed from prime numbers, people mainly focus on tabulating the prime knots. Unfortunately for Tait, there were warning signs on the horizon for Lord Kelvin's atom vortex theory. Mendeleev's first periodic table was published in 1869. The Michelson-Morley experiment sowed seeds of doubt about the ether in 1887, and the most damning result was the discovery of the electron by JJ Thomson in 1897. So there were particles smaller than the atom they were within atoms. But Tait was already in too much knots to stop. He had even ensnared his academic rival and close friend James Clerk Maxwell, famous for Maxwell's equations.

Thanks to Tait's influence, Maxwell became a knot enthusiast for the rest of his

life

. His last poem, written shortly before his death from stomach cancer, even begins: "My soul is an amphichiral knot, 'on a liquid vortex wrought.'" Aided in part by hundreds of letters with Maxwell, Tait published his list of knots. with up to seven crosses in 1877, the first mathematics work with the word knots in its title. Tait then stopped his search for seven years, stating in a speech: "The work necessary increases extremely rapidly" as the number increases. of crosses". free time needed "should try to expand this list, "if possible, up to 11." Two mathematicians came to his call for help, Thomas Kirkman and Charles Little.

Together, the three were able to find the 21 eight crossed knots, the 49 nine cross knots and the 166 10 cross knots in 1899, just two years before Tait's death, all of this was painstakingly done by hand. Tait admits in his article: "I cannot be absolutely sure that all such groups are essentially. different, one from another." In the process of tabulating knots, he had discovered the central problem of how to distinguish them. Somewhat miraculously, Tait, Kirkman and Little had done a nearly perfect job with their knot tables. Their list remained for 75 years without change, until a single correction in 1973. But more on that later.

For decades after Tait's death, little progress was made on the problem of knot equivalence. But in 1927, the German mathematician Kurt Reidemeister. proved a radical theorem. You only need three types of movements to transform two identical knots into each other. The twist, the push, and the slide, where you move a rope from one side of a crossing to the other. If you can show that they are connected by Reidemeister moves, you have shown that they must be identical. But we still don't know how to prove that the knots are different from each other because you could do Reidemeister moves on one knot for centuries without it ever looking like the other. knot.

And maybe they really are different, but maybe they are the same and you never took the right step to prove that to be true. It is possible that this is where Turing came from when he called the knot equivalence problem potentially undecidable. But in 1961, mathematician Wolfgang Haken created a computer algorithm that definitively solved the non-equivalence problem for this specific case of distinguishing any knot from a non-knot. That said, his paper was over 130 pages long and the algorithm would have taken longer than the age of the universe to run at large knots. In 2001, building on Haken's work, mathematicians found a way to distinguish between any knot and unknot by simply setting an upper limit on the number of Reidemeister moves required to connect them.

If you check all of Reidemeister's movement sequences up to that number, you can test whether the knot is untied or not. There is only one problem. That upper limit was two to the power of 100 billion n moves. As of today, the upper limit has improved dramatically to just 236 n to the power of 11. Now, although smaller than before, verifying all possible sequences of Reidemeister moves up to this number is still unfathomable. For a single crossing knot, this is greater than the number of stars in the observable universe. In 2011, mathematicians found an upper limit on the number of Reidemeister moves needed to connect any two knots or links, solving the entire knot equivalence problem.

This is the upper limit. First, raise two to the second power, then raise it again to the power of two. This operation is called tetration and it grows rapidly. Now continue doing this until you have raised two to itself 10 to the million n times. Close it with n again. This is easily the largest number we have ever shown in a video. But even just having a solution is notable, given that Turing thought the problem was potentially undecidable just 60 years earlier. If it is so difficult to distinguish two knots, how have we managed to tabulate 350 million different knots?

Well, there are some properties of a knot that never change, no matter how much you twist or tangle it. These are called invariants, and these invariants will be different for some nodes compared to others. So you can use them as a seal for a particular knot. They do not discriminate perfectly. I mean, some knots will share invariants, but if two knots have different invariants then you know for sure that they are different. The crossed number is itself an invariant. Two nodes cannot be identical if they have different crossing numbers. But the number of crossings is surprisingly difficult to calculate.

You can put additional crosses on any knot, such as just giving it a bunch of twists. Different variations of the same knot are known as different projections of that knot. The number of crossings measures the smallest number of crossings a knot can have, but it only works for the simplest projection of a knot, also known as its reduced form, but it is difficult to guarantee that a knot is completely reduced. Instead, we can use another invariant, one that is immediately true for all projections of a knot. Therefore, it will give the same value for both an unordered clover and a reduced clover.

This first invariant is tricolorability or whether a knot can be colored with three colors. Take a diagram of a knot and color each individual segment. These are simply separated by crosses below where you would lift the pen off the page. Tricolorability only has two rules. First, you should use at least two colors because you can color any knot with just one color. And secondly, in crossings, the three threads that cross must all be the same color or different colors. Basically there are no two-color crosses. There are only two categories of this invariant: either a knot is tricolor or it is not.

Identical knots must match. So if one knot is tricolor and the other is not, then you know they are different knots. It's hard to believe that tricolorability is constant in any possible projection of the same knot, but since you only need Reidemeister movesto move between projections, we just need to show that it is not affected by Reidemeister's movements. The turn is easy. Everything is now one color and it remains that way. With poke, the intersection of two colors means that the loop formed must become the third color. So we have three colors at each intersection. With the slide, you never have to break tricolorability because you start with three colors at three intersections and then change one intersection to a single color.

Therefore, any knot will maintain its tricolority, no matter what Reidemeister moves you make. This is a good time to point out that we never actually proved that trefoil and unknotting were two different knots, but now we can do that with tricolorability. The untied knot is not tricolor, since you can't use at least two colors to color it, and the shamrock is easily tricolor, just color each of the three segments a different color. All crosses have three colors, so it is tricolorable. We now know that all possible projections of the clover are tricolor, while all possible projections of the knot are not.

So these two knots must be different knots. This invariant is not very specific. It only offers you two categories in all knots. In fact, the next knot after the cloverleaf, the figure-eight knot, is not tricolor. There is always a crossover with two colors. So how do we show that this is different from the knotless knot, which is also not tricolor? Tricolorability expands to a much more powerful invariant called p-colorability, where p can be any prime number other than two. Instead of using colors, we will number each strand with integers between zero and p minus one. p-colorability has two rules.

First, you must use at least two different numbers. Secondly, in crossovers, the two bottom strands added and divided by p must give the same remainder as twice the top strand divided by p. Tricolorability was just a simple version of this. If we go from three to five colorability for the figure-eight knot, we can number the threads zero, one, and then this thread must give a remainder of zero, that is, four, and this thread must give a remainder of two, that is say, three. This knot has five colors, so it is not the unknotted one. p-colorability is a huge tool.

The unknot is completely colorless, so any knot with any colorability cannot be the unknot. p-colorability still doesn't cover everything. Some of the most powerful invariants at the moment, the ones that can distinguish between the most unique knots, are polynomials. The Alexander polynomial was the first discovered in 1923, even before Reidemeister moved on. Like p-colorability, it is based on only two rules. The first is that the Alexander polynomial of unknotting is equal to one. The second is that you can approach any junction of a knot and vary it in three possible positions: forward, back and apart. The Alexander polynomial gives a relationship between the three resulting knots.

Let's do an example. What is the Alexander polynomial for decoupling? Well, if we expand this separate junction and then vary it, we see that the other two knots formed are both unknots, so we can connect one for both, and we get that the Alexander polynomial for unknotting must be zero. Then we can do the same with the Hopf link. Taking this junction as the forward junction, we then see that the backward junction gives us the detour, and the separate junction is the unknot. So the Alexander polynomial for the Hopf bond is minus t to the power of 1/2 plus t to the power of negative 1/2.

And now we can make the clover. When we vary this crossing, we get that the backward crossing gives us the unknotting and the separate crossing gives us the Hopf link. So the Alexander polynomial is t minus one plus t to the negative one. The polynomial is designed so that we get separate results for as many nodes and links as we can, and this is recursive. We can compute the polynomial forever for increasingly larger knots. The Alexander polynomial remained unchanged for over 60 years as the knot invariant of choice. But in 1984, everything was turned upside down by an unlikely discovery.

Mathematician Vaughan Jones had been working on a type of algebra for statistical mechanics, a concept in physics, when he realized that his work resembled a series of knot theory equations. He traveled to New York to consult knot theorist Joan Birman at Columbia University, who helped him refine his equations into a knot invariant. They met again a week later and tested it with knot diagrams from Birman's binder, and quickly realized that Jones had discovered a new polynomial invariant. He scribbled all of his work into a 15-page letter. The Jones polynomial is like the Alexander polynomial, but with the more specific equation for the second rule that allows you to distinguish many more knots.

For this discovery, Jones won the field medal in 1990. The first new polynomial invariant sparked fervor in knot theory. Just a few months after Joan's result, six mathematicians each independently found an improved version of her polynomial with two variables instead of one. The editors of the American Math Society published all their papers together and called it a HOMFLY polynomial. Two Polish mathematicians missed the news and discovered it again a couple of months later, becoming the HOMFLY-PT polynomial. None of these invariants works alone. Just like if you were searching for a person, you would start by checking the first name, then the last name, then the date of birth, etc., to finally narrow the search down to a single person.

Similarly, knots have dozens of invariants that together uniquely identify them. With invariants to show if the knots are different and Reidemeister moves to show if the knots are the same, you can attack from two angles to accomplish the mammoth task of distinguishing each knot. But this method is not perfect. These two knots were listed side by side on Tait's knot charts for over 75 years. They were the same according to all invariant accounts. So, Tait and Little probably tried Reidemeister's moves to see if they could transform one into the other. And once they failed, they listed them as two separate knots.

Kenneth Perko, a lawyer who had studied knot theory, discovered them in 1973 while looking at Little's table of 10 cross knots. Suspicious of their similarities, he took out a yellow legal pad to sketch out some Reidemeister moves and quickly found a way to connect the two knots. These two projections now known as the Perko pair are the same knot. So Tait, Kirkman and Little's knot tables received their only correction. And instead of 166 ten crossing knots, there are 165. It took decades of work to tabulate the top 249 knots down to 10 crossings. No one dared to face the 11 crossings until John Conway.

He found all 552 and claimed that he did it in a single afternoon. This was the last tabulation done by hand. In the 1980s, Dowker and Thistlethwaite built a computer algorithm to count the 12 and 13 cross knots. Thistlethwaite later joined forces with Hoste and Weeks to tabulate the 14, 15, and 16 cross knots in a paper titled "The First 1,701,936 Knots." The method they used is still the one used today: they use a computer to enumerate all possible knots and then use invariants to eliminate duplicates. They split into two teams and checked their results, lining up perfectly on all but four knots on their first try.

In 2020, mathematician Ben Burton single-handedly tabulated the 17, 18, and 19 cross knots, bringing the total number of known prime knots to 352,152,252. His project was so computationally intensive that several hundred computers had to run for months before getting the final number. The most difficult part of knot tabulation is counting each knot and then carefully removing duplicates. But if you just want to generate a lot of different knots, you can make alternating knots, knots with crosses alternating above and below. This calculation is much simpler, although it omits most of the knots. And in 2007, this method was used to find alternating knots up to an absurd number of 24 crossings.

In total we know 159,965,097,353 knots. Of course, we are missing many things in the middle. Knot theory has always been pure mathematics. All algorithms, invariants and tabulations were knowledge for knowledge's sake. But in 1989, chemist Jean-Pierre Sauvage linked molecules around copper ions to form the first synthetic knotted molecule. This cloverleaf knot prevented the atoms from unfolding, trapping them in higher energy states to give the molecule new properties. Any type of knot tied in a molecule will change its properties, and we know of more than 159 billion knots. So if you can attach one molecule to each of those knots, that's 159 billion unique new materials created from a single molecule.

Although after clover, chemists have only managed to tie five other molecular knots to date. It is a difficult task. Since they can't simply push individual ions into place, the molecules must be built to self-assemble into knots. Knot theory helps identify which knots match the available molecular templates, symmetrical knots are easier for one, and how to arrange them to assemble the knot. The most complex knot created so far is knot 819 with 192 atoms tied around a central chloride ion. This molecule holds the Guinness record for the world's tightest knot, defined as the largest number of crossings per unit of length, in this case, eight crossings in 20 nanometers.

Since it is knotted around a chloride ion, once the ion is removed, this molecule is one of the strongest chloride binders there is. The field is still new for specific applications. Chemists simply focus on creating molecular knots before thinking about developing materials, but they hope to eventually build things like fabrics that are durable and stronger than Kevlar. Knot theory is also fundamental to biological processes that have saved millions of lives. Bacterial DNA consists of a single loop of the double-helix molecule. This shape means that it always forms a knotted bond when it replicates, and bacteria cannot separate into two cells with their DNA tangled in this way.

They have an enzyme called topoisomerase type two, which cuts and reconnects DNA. This turns their linked DNA back into an unlinked DNA so they can replicate cleanly. If you inhibit type two tophi summaries, the bacteria cannot replicate properly and actually die. This is how some of the most common antibiotics in the world called quinolones work. Human DNA, although not circular, is long enough to become tangled as well. Every cell in

your

body contains two meters of DNA. That's the equivalent of putting 200 kilometers of fishing line in a basketball. When this mess inevitably gets tangled, human type two topoisomerases come in to cross-exchange.

The human version of the enzyme is different enough from the bacterial version that antibiotics do not affect it. But human topoisomerases are sometimes intentionally inhibited. This stops replication and kills cells, predominantly rapidly dividing cancer cells. It is one of the most common forms of chemotherapy. Biologists needed knot theory to first understand the mechanism of type two topoisomerase. Once they noticed that the number of knots crossing the DNA in pairs was decreasing, they realized that it had to be cutting and reattaching entire double strands of DNA. And there are many other little-known topoisomerases that act on DNA.

Knot theory is used to analyze the knots that tie or untie and how they operate as a result. It's not just the DNA that gets knotted. 1% of all proteins have several nodes in their fundamental structure. If they break, they work poorly. Therefore, being able to distinguish the knots accurately helps understand the mechanisms of these proteins, as well as how to potentially repair or use them. When it comes to shoelaces, both common ways to tie the knot are made up of two clovers on top of each other. I'm going to tie a rope around my leg to make this easier to see.

When you go around the loop counterclockwise, you then form two identical clovers on top of each other. This is also known as a granny knot. But when you rotate clockwise around the loop, you get mirrored clovers on top of each other. This is also known as a square knot and does not loosen as easily. So we should all tie our shoelaces like this, clockwise, around the loop. Most of us are not. I mean, I'm usually not. I usually do it like this. A simple overhand knot is just the shamrock. The bowline knot, the most common knot for sailing or just holding things together, is the six-two knot.

And any knot made without using the ends, also known as inthe bite, which is just an unknotting. So a slipknot is an example of untying. In 2007, researchers Dorian Raymer and Douglas Smith conducted 3,415 rope-spinning tests in boxes to study how knots form in the real world. They ended up creating 120 different types of knots, some as complicated as 11 crosses. They found that a longer stirring time led to a greater likelihood of knots forming. Longer ropes did as well, except this probability decreased once the rope was placed in a smaller box, restricting its movement. So if you want to prevent something like headphones from getting tangled in your pocket where you can't adjust the string length or shake time, then your best bet is to limit them to as small a space as possible.

Raymer and Smith also proposed a model for knot formation in the real world. A series of loops are first formed when a rope is placed in a container. Then, when shaken, a free end of the rope is woven up and down through the loops, braiding into them to form knots. And let's see, a knot. So coiling the cables is actually setting you up for failure because you're forming a bunch of loops so that a loose end will braid neatly into a knot. So what you want to do is restrict its movement, either by using a small box or by increasing the stiffness of the rope.

DNA increases its rigidity by super-coiling, and you can do the same with your cables. I just fold it like this and then twist it from the middle. And this will stiffen the length of the cable. Naturally, this will want to curl up on itself and look like a big mess, but all you have to do is take the opposite ends of the earbuds and separate them, and there will be no tangles, no. knots. His study won an Ig Nobel Prize. It has been cited in studies about knots in surgical catheters and has even been linked to an Apple patent for stiffer headphone cables.

Knot theory began as a failed theory of everything and, for the next century, was an independent field of mathematics driven by nothing more than intellectual curiosity. But in recent years it has recovered its original potential. Today, knot theory is a theory that encompasses everything from headphone tangles to materials science and chemotherapy. In 1889, Kelvin gave a presidential address to the British Institution of Electrical Engineers about his failed atomic theory of knots. "I am afraid I must conclude by saying that the difficulties are so great in the way of forming anything resembling an integral theory that we cannot even imagine a finger pointing a path toward explanation." But this time next year, "this time 10 years", this time 100 years, "I cannot doubt" that these things that seem so mysterious to us now "will not be mysteries at all, "that the scales will fall from our eyes, "that we will learn to look at things in a different way" when what is now a difficulty "will be the only way to see the subject with common sense" and intelligible." Knot theory is a perfect example of how knowledge in one area can become a tool for understanding many others.

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