Where the Wild Things Are:
An Interview with Ken Millett

By Margaret Wertheim

This interview was first published in:
Cabinet, Issue 20, Winter 2005

MW: How did mathematicians become interested in knot theory?

KM: My understanding is that the mathematical origins came roughly with the studies of Carl Friedrich Gauss (1777-1855) involving electromagnetism and what would happen when an electric current is flowing through a wire. That would depend upon the configuration of the wire in space. That led later to the work in England of Lord Kelvin and Peter Guthrie Tait at the end of the nineteenth century. They were really the first to set about classifying configurations of knots, in part because they believed that atoms might be knots in the ether.

MW: Why did anybody think knot theory had application to atoms?

KM: At the end of the nineteenth century, scientists realized the world of knots was a discrete world, that is, different knot types could be distinguished from one another. This raised the question of whether knots were a good model for atoms, which seemed to have a similar character - there are hydrogen atoms and oxygen atoms and carbon atoms and so on, each with its own characteristics. In fact there’s a continuing interest in this kind of discussion today with string theory that also relates basic particles to complex knot-like configuration.

MW: How do mathematicians go about formalizing something like tying bits of string?

KM: Because it’s a human endeavor, the initial efforts were kind of intuitive. In order to capture knotting in a piece of string for example, it’s necessary to take the two ends and bind them together so that you have a closed system. If you don’t do that, you could move the knot off the end, making it disappear as it were. The first questions that arise for mathematicians are basic questions, such as: What does it mean to think of two knots as the same or different? That’s a fundamental question that would separate notions of topology from geometry, and you would get a different answer depending on what kinds of equality or similarity or equivalence you chose. Once you have that, then the question comes as to the size of the population: How many different knots are there? How would you distinguish between two instances of the same knot in different forms? The mathematics comes with trying to create a system and a process for identifying whether or not two knots are equivalent. And also from the desire to establish a complete classification, much as you would do with animals or other things in science. Mathematicians want a complete classification and we’re still struggling with ways in which to do that effectively.

MW: The population of possible knots is infinite, but how far have mathematicians gone in terms of classifying them? And how do you gauge how far you’ve got?

KM: In mathematics, the issue of “how far along you might be” is a very sophisticated question. We know lots of different ways of organizing information about knots. Since it’s a multidimensional enterprise, it’s hard to say how far we’ve got because from some perspectives we’re doing quite well, but if you change the question a little bit, then we’re not doing so well. If you organize knots, as was done classically, in terms of their pictures or diagrams, and you judge complexity in terms of these diagrams, then we’re not doing well. The classical way of studying knots is to project an image of each knot onto a wall, which reduces it to a planar diagram. When you do that, you see a bunch of under- and over-crossings of string within the knot. Now mathematicians say we should look at the picture with the fewest number of these crossings, what we call the “crossing number.” Looking at knots this way, we’re up to seventeen crossings. Even at this level there are enormous numbers of possible knots—(for seventeen crossings, we believe the number to be 8,053,249) The computational task of going beyond that is formidable, and I’d have to say that methods we have for distinguishing different knots at this level are really quite primitive. As you increase the number of crossings, it becomes exponentially more complex and at some point becomes computationally intractable.

MW: Can you explain how mathematicians try to distinguish between different knots?

KM: Aside from the crossing number, which is one feature, another measure is to look at how you can model a knot with the smallest number of line segments or edges. This is called polygonal modeling. You can think of this as taking a segmented ruler, like a carpenter’s rule, and bending it around to form a knot. Now we can ask: Can you model a knot from four edges? It turns out you can’t. The simplest knot is the trefoil or cloverleaf knot, and it takes six edges to model that. All knots can be modeled from some number of edges—this is what a lot of my work has been about recently—and the question is, “How many edges do you need to represent any particular knot?” That’s another measure. Another one is to imagine that your knot is inside a sphere. For mathematicians that’s not a very exotic task. Now take the “complement” of the knot—that is the part of the space that’s not the knot. In effect you’ve hollowed out the core of the sphere and created a knot-shaped hole in it. For all except a well-understood family of knots, this complement space can be given a hyperbolic structure and associated with that is a hyperbolic volume. This volume is another measure of the knot.

I should also mention another approach I’m partly responsible for. There are certain algebraic or polynomial equations that we call “knot invariants” that are associated with each different knot. In knot tables, you’ll see lists of numbers that define these equations. For many knots this is the main way we have of characterizing them. It’s a pretty chaotic system with all these different properties—there are even more than the ones I’ve just described—and it would be good if we could identify which are the core properties from which the others derive.

MW: There are knots that are “wild” and others that are “tame.” Can you tell us what a wild knot is?

KM: I would explain it by giving you an example and then saying, “Think like that.” Imagine I take a bunch of beads and we’re going to make a sort of necklace. Now take the first bead and I tie a little trefoil knot inside it. Now next to that I attach another bead, but this time with half the radius and inside that I make a smaller trefoil knot. Then next to that I put yet another bead with half the radius still. Now imagine I take an infinite sequence of these beads, each one half the size of the one before, and each with a tiny trefoil knot inside. At the end I wrap the string around so it’s closed. What makes this a wild knot is that it’s composed on an infinite sequence. This is actually one of the simplest examples you could make. Another way of describing a wild knot is that it can only be modeled by an infinite number of line edges. The opposite of wild is tame, which means you can build it from a finite number of edges, like the trefoil knot I mentioned before. Wild means you can’t.

MW: Do wild knots exist in the real world?

KM: In the physical science world, wild knots never happen. The reason is there’s a smallest scale at which you can do things. For wild knots, you have to be able to go smaller and smaller, until it’s infinitesimally small. But here’s another question: A knot is a one-dimensional structure; do such things exist as two-dimensional structures? It turns out the answer is yes. We can have what’s called a “wild sphere.” There was confusion about this. The mathematician who first wrote about it, James Waddell Alexander, said, “No, it’s not possible.” Then later, in 1924, he published a rebuttal saying here’s an example—which is now called Alexander’s Horned Sphere. Just as a wild knot cannot be modeled with a finite number of line segments, Alexander’s Horned Sphere cannot be constructed from gluing together finitely many triangles. It cannot be triangulated—it’s an infinite polyhedral object.

MW: Do wild things occur in all dimensions?

KM: I am embarrassed to say I don’t know for sure. I suspect the answer is yes. I have some recollection that strange things happen in higher dimensional space that might make things easier.

But let me tell you about another really charming problem. Imagine that we’re going to make a knot out of a polygon with, say, thirty-two edges. And we’re interested in what are the different knot types you can make with thirty-two edges. There’s a kind of space of them, a world of them, a population. And you can sample from them, sort of interview each one and ask, “What kind of a knot are you?” This is a very classical statistical kind of problem: You have a population and want to identify the constituents. We know there are a finite number of knots you can make with thirty-two edges; what we don’t know is whether that number is 600, or 1,000, or 10,000. There is no mathematical theory yet that’s even been able to give us a meaningful estimate.

MW: You must have an answer up to a certain number of edges?

KM: Eight. That’s all we know about right now. It turns out nine is already too complicated. And even for eight edges, there’s one case we don’t know for sure.

MW: For eight edges how many knots are there?

KM: For eight equal-length edges, there are eight or nine different knots, apart from the “trivial knot,” and we don’t even know whether it’s eight or nine.

MW: It seems extraordinary that something so simple could be so difficult.

KM: It is humbling.

MW: Can you explain the idea that knots can be seen like numbers—you can add them and subtract them, and there are even the knot equivalents of prime numbers.

KM: That is a theorem that goes back to the 1960s. Remember how we can construct a wild knot by making knots inside little beads? Well, this method also tells you how to combine knots. If you take one knot and put it inside a bead, and a second knot and put it inside a bead, and so on to make a little necklace, that process with some generalization also shows you how to take a knot and reduce it into its atomic pieces—its indivisible parts. The way it works is this: take a knot and let us try to find a sphere that intersects with it at two points, so that inside the sphere is one piece of the knot and outside is another. The technique tells us that given any knot, however complex, we can find a finite collection of spheres so that inside each sphere there is a piece of the knot that cannot be broken down any further into simpler pieces. You can move the spheres around so it looks like a chain of beads, each with a little knot inside. Every knot is composed of a unique set of these “irreducible” knots. It’s exactly like building up composite numbers from prime numbers. These irreducible knots are the primes of the knot world.

The unique factorization of knots is analogous to the unique factorization of integers. With numbers, we call it multiplication, but with knots we call it addition. One interesting thing about knots is that you can never cancel out a knot by adding it to another knot.

MW: So there’s no knot equivalent of negative numbers?

KM: No there’s not. But it’s more accurate to think in terms of reciprocals: If you take the number 2, then its reciprocal is 1/2, and if you multiply 2 and 1/2 together, you get 1. The knot equivalent of 1 is the trivial knot, or what is also called the “unknot.” But you can never cancel out any knot and get back to the unknot by adding it to another knot. There is no such thing as knot reciprocals.