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[ intro ]

Tying our shoelaces is one of those lifelong  skills we’re all proud to master as kids. It turns out that that simple act has a surprisingly deep history.

Early humans learned to tie knots before  they invented the axe, or even the wheel. And, about five hundred years ago, the Inca are believed to have used knotted ropes  as a way of storing huge amounts of information. Today, knots are everywhere you  look: in the ribbons around gifts, in the rope netting of children’s playgrounds,  and all over traditional boats and ships.

And they can even be in some unexpected places too, like the ornately carved Celtic braids that decorate old buildings, and the cable ties keeping Mars rovers neat and tidy. Of course, sometimes they are too everywhere. Like, when you’re just trying to get your  headphones out of your pocket, darn it.

Fortunately, knots are more than  just wire-based mortal enemies. By understanding how knots really work, we can use them as a tool to explore  some important problems in science today. That’s the goal of so-called knot theory, a  relatively new branch of mathematics which—you guessed it—is all about the  surprisingly complex study of knots.

It’s worth mentioning at this point that  a mathematician’s definition of a knot is probably different from what yours or mine is. Unlike our shoelaces, or the rope hitching a boat to the dock, a mathematical knot has no loose ends. The string can have crossings and intertwinings, but there can’t be any break. It’s one continuous loop.

To make a mathematical knot, imagine looping a string around a box, and tying a loose half knot across the top. Then take your two loose ends and fuse them together. Throw away the box and you’re left with a continuous loop of string that’s entangled in the middle.

Your very own mathematical knot! Of course, you could save  yourself the effort and go out and buy a pretzel. When the two ends are joined, that’s another mathematical knot.

Both of these are simple examples, but there are lots of ways that knots can get more complicated. Back in 1867, the scientist Lord Kelvin even suggested that the various chemical elements were actually different kinds of tiny  knots suspended in the mysterious ether. Like, hydrogen would be one kind of tangle, oxygen would be another, and so on.

It’s a nice idea, but a bad guess. There is no ether, and atoms turned out  to look nothing like knotted strings. But Kelvin’s imaginative suggestion did kick-start a new kind of thinking about knots, one that was rooted in mathematics rather than aesthetics.

Knot theory was born. Compared to some other branches of  mathematics, like algebra and geometry, which have been studied for thousands of years, the math of knots is only a couple centuries old. Although things do get pretty complicated, thinking about the shape of knots is actually something you can do without  always needing advanced math.

Like, you don’t have to be great at calculus to get your head around the problem. And lots of breakthroughs in knot theory have come from researchers just thinking about it in their own unique way. More on that in a moment, but, first, let’s take a look at how all this  knotty-ness is actually being used.

Take biology, for example. Almost every cell in your body  contains around two meters of DNA, packed into a space less than one  hundredth of a millimeter across. All that DNA has to be coiled up in a way that not only fits inside a cell’s nucleus, but is also accessible.

After all, it provides the instructions for all other processes in the cell. But just like those headphones in your pocket, long strands of DNA are just dying to get tangled, and science has shown that the longer the  strand, the more likely it is to get twisted up. So biologists are looking to knot theory to  see how DNA manages to always stay ordered.

The DNA in bacteria is a particularly good target, because— just like in the mathematical  ideal—bacterial DNA is usually arranged in a single continuous loop. DNA’s famous, twisting  double-helix structure means that, when a bacterium replicates its genetic material during reproduction, it inevitably creates knots that would  kill the microbe if left unchecked. But, biologists have discovered a special  kind of enzyme that finds these tangles, snips open the loop, straightens things  out, and seals everything up again.

Knot theory helps biologists understand how   long it will take the enzymes to do their job on various kinds of twists. And by making antibiotics that target these knot-busting enzymes, medical researchers are seeking to harness tangles in the fight against infection. Scientists are also developing new and interesting  materials made from knotted nanoparticles.

Synthesising tiny knots is  tricky business, but, in 2017, chemists created what they call the ‘octofoil knot’ out of just a hundred and ninety-two [192] atoms. They still aren’t sure of its properties, but hope it will be useful for creating things like ultra-light armor or flexible surgical sutures. But for scientists and engineers to truly harness the power of knots, they need to know more about them.

That’s where math comes in. The first challenge for mathematicians is to try and figure out how many different  types of knots there actually are. One way to classify them is by how  many crossings a given knot has.

That is, how many times the string  passes over or under itself. Some knots can look really complicated, but can actually be untangled to be much simpler. The key is to find the minimum number of  crossings inherent in a given arrangement.

The simplest kind of knot is actually something most of us wouldn’t consider a knot at  all—just a closed loop with no crossings. Mathematicians call it the trivial knot, or the unknot. You might know it as a circle.

Because of the way a looped string works, there are no knots at all that  have only one or two crossings. There’s one knot with three  crossings, called the trefoil knot, and one with four, called the figure eight. So far, so simple.

But from five crossings upwards, things get more complicated. There are two different knots with five crossings. No matter how much you move them around, it’s impossible to turn one five-crossing knot into the other.

Similarly, there are three unique knots with six crossings, seven knots with seven crossings, and, after that, the numbers start getting out of hand. By the time you get to 10 crossings, there are 165 completely different, unique knots that can’t be simplified any further. To try and keep track of them, mathematicians in the late 19th century tried creating tables of the unique, or so-called prime knots.

These tables have been an invaluable tool in understanding more complex knots, since some tangles with many crossings can actually be broken down into a collection of more simple knots. Those are called composite knots, and just like large numbers can be broken down into their prime factors, large composite knots  can be broken down into their prime knots. Unfortunately, it’s not  always that straightforward.

It can actually be really hard to figure out how many unique knots there are, and if the knot you happen to be thinking about has already been found. This ambiguity is especially troublesome  as the knots get more complex. Like, in 1974, a mathematician found an error in the knot tables that had been around for nearly 100 years.

Two of the unique knots listed with 10 crossings were actually duplicates of one another, and no one had noticed for all that time. I, personally, don’t actually find that that surprising. How do you keep all this in your head?

It’s such a tricky problem that even computers haven’t been able to solve it completely yet. Knot theorists have developed a bunch of other ways to classify knots with algorithms, equations, or geometric descriptions. One way is to flatten out a knot and imagine it as a complicated roadway with overpasses and underpasses.

You can then describe the route numerically, writing which underpass matches each overpass. Other, more involved techniques relate the numbers of different types of crossings to each other in an equation to find out if knots are equivalent. And there is even an approach that involves trying to color knots, to help group them and tell them apart.

All of these tools have helped mathematicians over the years, but the problem is, none of them are perfect. If any one of these approaches reveals that your knot is different to others you’re comparing it with, then great! You might have found a new knot.

But if they say it’s equivalent, you can’t be 100% sure that that’s the case. Sometimes one mathematical solution can describe several different knots. So that particular method isn’t enough to distinguish them.

And, if that wasn’t complicated enough, you can always add extra dimensions to the mix. In 4D, so the three dimensions of space and one dimension of time, instead of knotting loops of 2D string,  mathematicians consider hollow knotted spheres. If that hurts your brain a little, join the club.

This is some seriously mind-bending stuff. Mathematicians study these knotted spheres by considering what they would look like if  they sliced through them at various points. A slice through an unknotted  sphere would be the trivial knot - a simple loop. But if you slice through a knotted sphere, you could end up with one of  the other known prime knots.

Figuring out ‘sliceness’—that is, whether a given knot can be produced by slicing up a 4D knotted sphere— is another way that theorists can try to distinguish between different knots. And that’s actually been done for all of the known knots with up to twelve crossings. One particular knot, with eleven crossings, had evaded sliceness classification for decades, until very recently.

The problem was finally solved by a grad student in 2020, after just a few days of considering it in her spare time. Four-dimensional knotted spheres are good fun for masochistic mathematicians, but they’re also helping physicists in their mission to understand the strange and unpredictable quantum world. Quantum particles don’t typically follow the shortest path from A to B; instead, they can take any number of strange loops and zigzags.

So, to try and predict the behavior of things on the quantum scale, physicists need to consider all the possible routes these particles could take. Knot theory, and its goal of reducing complex entanglements into simpler, more fundamental forms, is helping solve that problem. The numerical approaches to classifying knots, while not perfect from a mathematical perspective, are nevertheless helpful for predicting most likely quantum paths.

And that information helps physicists not  only understand the weird quantum world, but also how to harness it to build things like quantum computers. What’s more, once mathematicians have  conquered knots in four dimensions, they can add even more to help them picture the strange intersecting dimensions implied by string theory. Knots have been a part of human culture for almost as long as humans have been around, but it’s only now that we’re realizing their potential to make sense of the world.

So, next time you’re tying your shoelaces, try not to get overwhelmed. You’ve been doing this since you were five. Thanks for watching this episode of SciShow,  and thanks to our patrons for helping us make it happen.

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