Find a string. Really. Do it. Now twist, tie and tangle it as much as you like. Finally, attach the two loose ends of your string together to make a closed loop. (This is a crucial step.) What you hold in your hands is one of the most exciting mathematical objects of the 20th century: a knot. (Hopefully you didn’t use your headphone wire.)
Which knot in each of the following pairs can be unraveled into a circle without opening the loop?
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Any knot that can be unraveled into a circle, as two of the above can, is equivalent to the delightfully named unknot. But what about the other two knots? How could we possibly be sure that no amount of pulling, twisting and tying could turn them into circles without wearing out our headphone wires? Is there a way to know that they’re really two different knots—that they can’t be arranged to look identical to each other? These are some of the fundamental questions in the mathematical field of knot theory.
Humans have been tying knots since prehistoric times for their practicality and beauty. Mathematical knots, in particular—knots with attached ends—show up as a recurring motif in Chinese and Celtic artwork that dates back centuries.
But the mathematical classification of knots didn’t begin until the 1870s. Researchers who created early tables that categorized different knots were driven by the idea that atoms were knotted vortices. They reasoned that by tabulating all possible knots, they could make a periodic table of elements. These chemical efforts were spearheaded in part by Lord Kelvin (creator of the eponymous temperature scale). Even after chemists discovered that atoms aren’t actually knots, mathematicians ran with the idea, forming an entire branch of study around the tricky creatures.
Mathematicians typically manipulate knots through diagrams like these:
(But we do still pull out physical string now and again.) Each place where one strand passes over another in a knot diagram is called a crossing. An unknot (also known as the trivial knot) can be drawn with any number of crossings. Here are unknots with 7, 11 and 15:
What is the smallest number of crossings you can use to craft a knot that is not trivial? Draw it.
Any possible manipulation of a knot diagram that does not fundamentally change the knot can be achieved by a series of three operations called Reidemeister moves. They include taking a strand and adding or subtracting a twist, sliding one strand over or under another, and passing a strand under, over or in between two strands in a crossing:
Any characteristic of a knot that doesn’t change when rearranged in this way is called a “knot invariant.” Take, for example, tricolorability. A knot is tricolorable if each arc in a diagram of that knot can be assigned a different color according to the following rules:
At each crossing, all three segments are either the same color or different colors.
Multiple colors are used.
A standard unknot, represented as a circle, certainly isn’t tricolorable. There’s only one arc, so it’s impossible to satisfy rule #2. But what if we use different diagrams of the unknot, as above?
Using Reidemeister moves to represent all possible things you can do to a knot diagram, show that tricolorability really doesn’t depend on how you draw the diagram.
Show that the “trefoil knot” below is tricolorable and therefore not equivalent to the unknot.
Not all knots can be distinguished using tricolorability.
Show that no matter what coloring you use, this figure-eight knot, like the unknot, is not tricolorable:
For any given knot, there are only two options: either it is tricolorable or it isn’t. We say, then, that tricolorability assigns each knot a yes or no value. But more complex knot invariants can assign each knot a number, a polynomial or even a mathematical object called a group. The crossing number of a knot, for example, is the smallest number of crossings it can be drawn with. The unknotting number is the smallest number of crossing changes (rearrangements of strand order in a particular crossing, as shown below) needed to transform a knot into the unknot. The trefoil knot has unknotting number 1.
Take a guess at the unknotting number of the following knots:
The diagram on the right above shows a figure-eight knot connected to a cinquefoil in what is called a “connected sum.” It was long believed that the unknotting number of the connected sum of two knots was the sum of their individual unknotting numbers. In this case, that’s true—the unknotting number of the third knot is 1 + 2 = 3. But in a recent twist of events, it was shown this is not always the case.
A knot that cannot be described as a connected sum of two other (nontrivial) knots is called a “prime knot,” and, just as prime numbers are the building blocks of positive integers, these knots make up the structure of all other knots. Prime knots are listed in standard knot tables like this one inspired by early knot tabulators such as Lord Kelvin:
The cinquefoil we saw above is officially dubbed 51, which means its crossing number is 5 and it is the first one with that crossing number listed in the standard knot table. So far, mathematicians have managed to tabulate all prime knots of up to crossing number 20. (For a sense of scale, there are 1,847,319,428 prime knots with crossing number 20, excluding mirror images.)
The mathematics of knot theory can also be used to study links, which are interconnected knots. Instead of one loop knotted up in space, links can contain any number of knotted components linked together.
Here’s an example of a link whose components are all unknots:
This link is known as the “Borromean rings,” and it has an interesting property. All three of the loops are linked (that is, they can’t be separated out from the others without cutting), but no two components are linked together independently: removing any one of the rings leaves the remaining ones unlinked.
Can you find a four-component link with this same property? None of the four components can be separated out, but removing one will separate all the remaining ones.
The leap from knots to links is a relatively small one, but mathematicians can extend the ideas of knot theory to study mind-bending concepts such as higher-dimensional knots, surfaces with knotted edges and difficult-to-imagine objects obtained by subtracting a knot from three-dimensional space.
And although chemists set knots aside as a way to describe atoms, they now use knot theory to examine the structures of different molecules and synthesize new materials. Biologists use it, too, to understand the way proteins in our cells tangle up and to create effective gene-editing technologies. These are just more examples of how mathematics and the sciences play off of each other to help us better understand the inner workings of the universe. Happy knotting!