Everyone knows what a knot is. But knots have special significance in math and science because their properties can help unlock secrets hidden within topics ranging as widely as the biochemistry of DNA, the synthesis of new materials and the geometry of three-dimensional spaces. In this episode, host Steven Strogatz explores the mysteries of knots with his fellow mathematicians Colin Adams and Lisa Piccirillo.
Steven Strogatz (0:03): I’m Steve Strogatz, and this is The Joy of Why, a podcast from Quanta Magazine that takes you into some of the biggest unanswered questions in science and math today. In this episode, we’re going to be talking about knots.
We all know what knots are, right? They’re, like, the kind of knots that you tie in your shoelaces, or the kind that you use to secure your luggage to the top of your car. If you take a string, which has two free ends, and tie a knot in it, that kind of a knot can come undone. Sometimes the free ends will slip loose, and the knot will just unravel. But if you fuse the ends together, sort of glue them together, then the knot will be locked in there, trapped in the loop. Then the question becomes, can you somehow remove that knot in the loop, without cutting the string, by just cleverly manipulating the loop somehow or wiggling it?
Well, if you can, that’s not much of a knot at all. That’s just a circle, something equivalent to a simple loop, or what mathematicians dismiss as the “trivial knot.” But if you can’t undo it, well then, that raises all sorts of questions like, how far can you simplify a tangled loop?
How do mathematicians distinguish different types of knots? How many different kinds of knots are there? And why do mathematicians and scientists care about knots anyway? Turns out, there’s lots of real-world applications for this branch of math, now called knot theory. It started out with the mystery of the chemical elements about 150 years ago, which were, at the time, thought to be different kinds of knots tied in the ether. Nowadays, knot theory is helping us understand how enzymes can disentangle strands of linked DNA. And also, knot theory has potential in basic research to create new kinds of medicines, including some chemotherapy drugs. But in math itself, knot theory is helping mathematicians work out the riddles of higher-dimensional spaces.
Joining me now to help unravel some of the mysteries around knots is Colin Adams. He’s the Thomas T. Reed Professor of Mathematics at Williams College, and one of the managing editors of the Journal of Knot Theory and Its Ramifications. Adams also wrote the comic book Why Knot? Later on, we’ll be speaking with Lisa Piccirillo. She’s an assistant professor of math at MIT, and she recently solved a long-standing math puzzle about a knot called the Conway knot — actually when she was a graduate student. Colin Adams, thank you so much for joining us today.
Colin Adams (2:35): Oh, it’s really fun to be here. Thanks very much for inviting me.
Strogatz (2:38): Well, it’s a real treat for me, Colin, I mean, I’m a big fan of your work. I love your books about knots. I’ve learned so much from them. So okay, so I tried to give a quick introduction to knots and knot theory, but I’d love to hear how you would explain them to someone who’s never heard of them or thought of them outside of tying their shoelaces?
Adams (2:55): Yeah, I mean, it’s one of the great advantages of being in this field in mathematics is, you know, usually, if you sit next to somebody on a plane, and they say, “What do you do?” and you’re a mathematician, you’re going to have a very hard time explaining to them what you do. But I have this huge advantage that I can take my shoelace, and I can take it off my shoe, and I can pull it up, and I can say, okay, I can tie a knot in this shoelace, and then glue the two loose ends together, and then I can try to study that object and decide is it really knotted — as you were describing, Steve — is it really knotted? Or is it not? Could you disentangle it without cutting it open?
And it turns out, this is a very hard problem. I mean, you could imagine you have—somebody gives you a tangled-up mess of string with the two loose ends glued together. And they ask you the question, can you disentangle it? And you spend the next six years trying to disentangle it. And after six years, you have not succeeded. But you still don’t know that five more minutes of work might do the trick, right? So you’d really love to have some techniques, some mathematical techniques that will allow you to determine whether or not you’re going to ever succeed.
Strogatz (3:28): So, knots are all around us. Can you give us some other examples where they pop up in the real world?
Adams (4:00): Yeah, so I mean, knots have been around for a very long time. In terms of actual usefulness, in terms of science, the one example that you gave, which is a really good example, is the example of DNA, where DNA is inside the nucleus of a cell. And it’s all packed in there — people have described it as putting 100 kilometers of fishing line in a basketball. And so it’s this huge mess in there. And yet, it has to be able to do transcription, recombination; it has to be able to create copies of itself that can then be separated off of itself. And it turns out that that requires these enzymes, as you mentioned, these enzymes inside the nucleus of the cell, that will allow the enzyme to take two strands of DNA, pull one up right next to the other, cut the one open, push the other one through, and then close it up again, to make what’s called a knot theory of crossing change. And this is going on inside us all the time that these enzymes are actually doing that to the DNA. And if you stop the enzymes from doing that, that will prevent the DNA from recreating itself, which is in fact the reason why this is now in use in chemotherapy. There are these chemotherapy drugs that prevent the enzymes from acting on the DNA and changing the knotting and unknotting. So that’s an example of how knots appear in DNA.
Another example that I think is a very interesting one is for synthetic chemistry. So in synthetic chemistry, you’re trying to synthesize new molecules. And imagine you have a molecule made up of a set of atoms all bonded together, and it forms a cycle. And so that would be an example of a trivial knot that is actually on the molecular level. Now imagine taking that trivial knot, cutting it open, tying a knot on the molecular level, and then gluing the two loose ends together. And once you’ve done that, you have the same constituent atoms bonded in exactly the same order. Only now you have a new substance, a knotted substance, whereas the previous substance was unknotted.
So for each kind of knot that you can have — of which there are an unlimited number, but even for small crossing number, millions and millions — each one of those knots can now become a new substance. So the synthetic chemists are just salivating over this idea. And they’re working very hard to try to come up with ways to knot on the molecular level.
Strogatz (6:09): It sort of reminds me of being an undergraduate. I was taking an organic chemistry course during a time when my parents had put the heat on me to be a pre-med. And so I was taking orgo [organic chemistry] like every other pre-med, and I remember being really fascinated, as someone who likes math, that our professor talked about just what you described. But at that time, it was only hypothetical. They didn’t know how to knot or link molecules synthetically at the time, but he said it was interesting to think about. You could take, exactly as you say, you know, combinations of — could be a polymer with hydrogens and carbons and, and they could keep exactly the same atoms in the same arrangements, each with the same neighbors as before. But you could make an isomer that was a topological isomer because it had a knot tied in it. But you know, like, would that have different chemical properties or not?
Adams (6:58): Yeah, and the expectation is, in general, the answer is yes, that it should have different properties. And you know, you could have one that’s acting like an oil, and the other one is acting completely differently. And so — it’s interesting, the Navy actually funded a lot of research on this, and it really came about because they were very interested in coming up with new substances that they could coat their submarines with, so as to avoid radar. So it’s kind of interesting that that’s where a lot of the money came from.
Strogatz (7:22): No kidding.
Adams: In order to study this, yeah.
Strogatz: As like a kind of stealth technology.
Strogatz (7:27): Wow, topology in the service of stealth. Huh, I’d hadn’t heard about that. That’s cool.
We’re starting to see why people in fields like chemistry or the military, or doctors working with chemotherapy drugs would care about these kinds of issues of knots. But why are they so interesting mathematically?
Adams (7:46): Knot theory is a subfield of the — of the broader field of topology, topology being this area of mathematics where you treat objects as if they’re made of rubber. So for instance, you can have a sphere, which you consider equivalent to a cube because you could just pull the six corners out and transform a sphere into a cube without cutting or pasting. And you consider a doughnut, the surface of a doughnut, a torus equivalent — the famous example is equivalent to a coffee cup, because you could deform the surface of a coffee cup, kind of push up from the bottom, and deform it until it looked like the surface of a doughnut. So those are considered the same.
So topology is this very broad area of mathematics. Knot theory is actually a subfield of that. Because again, we’re treating these knots as sitting in space, but they’re treated as if they are actually rubber knots and you’re allowed to deform them. And so in understanding questions about knot theory, you’re actually asking questions that are — particular questions that also can be generalized to questions about topology.
Strogatz (8:50): So this is a really interesting distinction, an important one that you’re making, this idea of — the knots that mathematicians like to think about are of this kind of idealized, rubbery nature where they can be twisted and bent and deformed and stretched as much as we want. And actually, you know, like in the real world, when people think about knots — say the knot in your shoelaces. There’s other aspects that mathematicians tend to ignore, like real knots often rely on friction. So maybe you should tell us everything that you’re going to leave out in thinking about knots as a mathematician?
Adams (9:24): Yeah, no, that’s a great question. Yeah, knots are funny when you’re thinking about them as a mathematician. So for instance, one of the things you’re thinking about, if you’re thinking that your knot is infinitely thin, you’re thinking of it as a circle that has been cut open and tied in a knot. But that circle has a thickness of one point. So it’s a very skinny, thin thing that you’re actually considering when you consider knot theory. And then, as we said before, you’re going to cut it open, tie a knot in it, and glue the ends together. And then you’re allowed to deform it any way you want, but you’re not allowed to have it pass through itself at any time. That’s one of the rules that we’re using.
We have some other rules that are less obvious. Like, you can’t tie a knot in it and then tighten it up so it becomes smaller and smaller and smaller until it just becomes, as you can on your shoelace, get a little knot on your shoelace — you’re not allowed to do that in knot theory. And one of the ways you avoid that is you talk about it being what’s called piecewise linear. So you can imagine all of your knots are actually made from a bunch of little sticks that are glued end to end to make your knot, and you’re not allowed to increase the number of sticks without bounds, you have to keep the number of sticks to a finite number at all times. And that avoids this issue of having smaller and smaller knots within your, your knotted circle. So those are a couple of the conditions that we have to put on knots in order to talk about them mathematically.
Strogatz (10:44): Like, what kinds of things in math are being learned by studying knot theory? So what I’m looking for here is, what is it telling us about spaces, or about algebra? Or, you know — what are the connections within math that one gets by thinking about knot theory?
Adams (11:00): The most basic question somebody might ask you in knot theory is: They hand you a tangled-up mess with the ends glued together and they say, can this be disentangled without cutting it open? Is this the trivial knot? So you’d like to be able to tell whether or not a knot is the trivial knot.
More generally, somebody hands you two knots. And you’d like to be able to tell, are they the same knot? Are they the same knot, or are they different knots? And so this is again, another one of the fundamental questions. One of the problems that people have been working on since the 1880s is the tabulation of all of the knots, where you try to make a table of all the knots up to some level of complexity. In tabulating knots, we tabulate them by keeping track of the number of crossings and trying to determine how many knots have a certain number of crossings.
And by crossings I mean that if you take a picture of your knot, and you look at that picture, there are going to be places where the knot crosses over itself or under itself. And those are our places that we call crossings. And so we can figure out how many knots there are that have three crossings, and there’s only one, called the trefoil knot. The trivial knot is the only knot with zero crossings. And then you can have — there’s one four-crossing knot, there are two five-crossing knots, there’s three six-crossing knots, seven seven-crossing knots, and you can go up from there. And up to now, so far, we’ve tabulated all the possible knots up to 19 crossings. And there’s over 300 million of those knots up to 19 crossings. And so this is one of the things that people do, is they try to figure out what are the possibilities, and in particular, given two of those, how do you decide if they’re the same or not?
(12:33) Now, in order to do that, you create these quantities called invariants. And invariants are just quantities that you can associate to a picture of a knot. And if two different pictures represent the same knot, they should have the same quantity associated with them. That quantity can be a number; it can be a group if you’re in algebra; it could be some other kind of an object from mathematics, but it’s a way of distinguishing between knots.
And so for instance, one of the areas that I’ve worked on since I got my Ph.D. was on hyperbolic volume of knots, which turns out to be a very powerful way to distinguish knots. And the reason I mention that one is because, Steve, as you were asking before, how does knot theory tie into other areas of mathematics? And the hyperbolic volume of a knot actually comes out of hyperbolic geometry. So this is a geometric invariant. And even though knots themselves are objects that can be deformed, and I think of them as being made of rubber — I use the term topological because they’re rubber, they’re deformed easily — but it turns out that when you try to figure out the hyperbolic volume, it is a unique number that is associated to a knot that’s hyperbolic.
So for instance, the knot that has four crossings has a hyperbolic volume of 2.0298… out to arbitrarily many decimal places. And you can use that number to distinguish between the knots. But in order to understand that area of knot theory, you need to understand geometry, you need to understand algebra, because the group that’s associated to a knot, which is called the fundamental group actually is realized as a set of isometries of hyperbolic space, isometries being maps that preserve distances. And so you need to understand algebra, you need to understand some analysis, you need to understand some geometry. So it brings together a whole lot of different fields of mathematics, all with the purpose of understanding knots.
Strogatz (14:26): So invariant might not be the most common word in ordinary English, but we could think of it almost like a signature or a fingerprint or something.
Adams (14:33): And there’s a very famous example, actually, which is kind of an interesting story. Back in the 1890s, there was a mathematician at what was then called Nebraska State University named C. N. Little, and he was trying to tabulate all the knots of 10 crossings. And he made this list, and he made a list of 166 knots of 10 crossings. And then that list existed until 1975, I think it was.
And in 1975, there was an amateur mathematician from New York, a lawyer from New York named Ken Perko. And he was looking down the table of knots. And many of the knots had been distinguished using an invariant called the Alexander polynomial. And he noticed two of the knots had the same Alexander polynomial. So he made the one knot out of string, he rearranged it. And it turned out to be the same as the other knot. And so it turned out that table had been wrong for 75 years. And that, in fact, those two knots were the same knot. And that that pair of knots is still known as the Perko pair, even though it’s actually only one knot, but in the tables, they’re called the Perko pair.
Strogatz (15:40): I love that example of the Perko pair. I mean, that exactly summarizes what is so delicate and fascinating about the mathematical aspect of this subject. That, as you say, this example could have been just sort of lurking there in the table 70 years. Who knows how many eyes had looked at these and thought, yeah, that knot looks different from that knot. They both have 10 crossings, but I can’t see any way to make them the same. So nobody thought about it.
When you mentioned earlier that knot theory is related to all kinds of deep questions in geometry and topology, I think it’s probably a good time for us to now start talking about the space that surrounds the knot. Right? I mean, knot theory is not just about the loop. But it’s a loop sitting in three-dimensional space. Or in some cases, knot theorists find it helpful to think about things that are knotted in four-dimensional space. So let’s talk about some of that.
Adams (16:34): Yeah, no. So it’s kind of a fundamental question. Initially, in knot theory, everybody was thinking about knots living in three-dimensional space, because that’s the natural space that we all live in. But you can also think about knotted spheres in four-space, knotted three-dimensional spheres in five-space, knotted four-dimensional spheres in six-space.
And so you can go up — dimension, dimension, dimension. You can also think about knotted circles in four-space. And that’s a really interesting category. So for knotted circles in four-space, it turns out that every knotted circle in four-space can be unknotted. And so every knot is a trivial knot once you go up to four-space. I’ll take a knotted two-sphere in four-space rather than a knotted circle in four-space because all of knot theory kind of disintegrates when you go up to four dimensions because of the fact that every knot can be disentangled in four-space.
Strogatz (17:29): This idea that knots can untie themselves in four dimensions, I think it’s so fascinating. And if we could give people some intuition about it, I think that they’ll find it very memorable and use it as like a little cocktail party trick. So first of all, maybe you should say, we know how to think about three dimensions of space. But how are you going to think — because we can’t really picture a fourth spatial dimension — what is your fourth dimension going to be in this intuitive argument?
Adams (17:54): Yeah, so a model that I really like for describing four-dimensional space is, I use what’s called a color model. And so if you think about the color spectrum — you know, red, orange, yellow, green, blue, indigo — I can imagine making four-space out of a collection of three-dimensional slices of four-space, each of which has a different color. And it’s going to be a continuum of colors. So it’s going to go: Red will be one slice, and then a little more reddish orange, and then a little more reddish orange going over to orange. And I’ll have this continuum of slices, each slice having a unique color. And this is an advantageous way to think about four-space, because you’re not using time as your fourth dimension. So you can actually move around in this space, imagine what it would be like to live in this space. So it’s a really nice model for four-space.
Now. I want to ask the question: Can I show that a knot can actually be unknotted in four-space? And so I can imagine taking my knot in the green slice of four-space. And so this is a three-dimensional green slice of four-space, and I’ve got my knot sitting there. And then what I’m going to do is I’m going to push just a little bit of my knot into the neighboring yellow color, so it’s going to become kind of a greenish yellow. But in my three-dimensional green space, I won’t see that part anymore. It doesn’t exist in the green three-space. It only — that little part only exists in the neighboring colors that are nearby. And so in my green space, I now see a knot that’s been cut open. And I can disentangle it. And so I can disentangle it there, and then I now have my knot in four-space, and it’s completely disentangled. So in that sense, there are no nontrivial knots in four-space.
Strogatz (19:32): All right, there’s no knotted loops in two-space because there’s not enough room in the plane. And there are no knots in four-dimensional space, because there’s too much room. They can just trivially untie themselves by blushing when they need to cross through themselves, by changing their color at that, at that crossing point. So it’s this fantastic thing where if you go up by two dimensions, like a loop is a one-dimensional thing like a one-dimensional thread, you could say where you are on the thread by just giving one number, how far you are along the thread as you go around it, like one lap around. So you’ve got this one-dimensional thing sitting in three-dimensional space. And that’s where knotting happens. But we can jack the whole thing up by one in both, we could now go up to a two-dimensional object like the surface of a sphere, but let it live in four dimensions. So again, something two-dimensional, two dimensions higher. And you’re telling me that I can take a sphere and tie a knot in it, if it lives in a four-dimensional space.
Adams (20:35): Yeah. And so as an example, let’s see if I can explain — give an example of how you might do that. Let’s go back to the color model. So in the color model, remember, we had these three-dimensional slices, each a different color, and I’m going to try to create a knotted sphere. Okay, and here’s how I’m going to do it. I’m going to take a trefoil knot, which is just that knot with three crossings, I’m going to make a red one. And then I’m going to, right next to it, I’m going to have a reddish orange one and an orange one. And in all the continuum, I’m going to have all of these trefoil knots, okay, that are… So in each three-dimensional slice of four-space, I’m going to have one of these trefoil knots. But I’m going to make the ones as I get out towards the left end of the spectrum, the red end of the spectrum, I’m going to make them get a little bit smaller and smaller and smaller. And when I go out the other end, towards the indigo end, I’m going to make them get smaller and smaller and smaller. Until when I get to indigo, it’s just a point. And when I get to red it’s just a point.
Okay. And that object, believe it or not, is actually a sphere. Because you can cut up a sphere by starting at one end at a point and then slice it and you get a small circle and then a bigger circle and a bigger circle until you get to the — to the middle point, which is the biggest circle and then the circles shrink down to a point. Only now this time, all of those circles are knotted.
Strogatz (21:51): Oh my god, you’re my hero. I have lived a long time thinking about knots. And I never understood how you could have a knotted sphere in four dimensions. But let me try to say what you just said. I think I got it. This is cool. So you’re going to take a trefoil knot, that’s our little knot with three crossings, it looks like a cloverleaf, kind of, except it’s a three-leaf clover, but ignore that.
So I’ve got this trefoil knot. And you imagine a rainbow collection of clover leafs going from red at one end to indigo at the other end. I’ve got this whole spectrum, a whole rainbow of trefoils sitting next to each other. And then you say at either end, shrink them down so that at the at the very ends, the red one is just really kind of almost a point. And the indigo one is a point. And now I have something that you say reminds you of, if you looked at the globe, an ordinary sphere, and started doing slices from the north pole all the way to the south pole, taking slices along lines of latitude — that I would start by just clipping off the north pole, that’s a point. Then as I go down, I get bigger and bigger latitudinal circles. Then I reach the equator, I get a big circle. Then my circles start getting smaller again and then they shrink to a point.
So a sphere is made up of these — a point followed by increasing circles followed by shrinking circles followed by a point. And you just did that with the trefoil knot. Except that it happens that it’s got knots in it, but you’re saying somehow we don’t see that it’s still a sphere worth of trefoils.
Whoa. I love what you’ve just explained about thinking about knotted spheres in four dimensions. Because one of the things I want to talk about with my next guest, with Lisa Piccirillo, is work that she did on whether a certain kind of knot with 11 crossings, called the Conway knot, is “slice” — which was a very famous long-standing problem in knot theory. I mean, how should I think about slice?
Adams (23:51): Yeah, so slice comes back to this fact that I mentioned which is every knot in four-space is trivial. In other words, every knot in four-space can be deformed to the trivial knot, and once you deform it to the trivial knot, you could actually deform it into a plane, a two-dimensional plane in four-space. And that two-dimensional plane bounds a disk.
Okay? So that knot bounds a disk. And when you first look at the knot, before you’ve deformed it to look like just the trivial knot sitting in a plane, it could be really messy, ugly things sitting in space, and you can’t even tell that it actually is trivial, but we know it is. And therefore there is this disc that is just bent way out of shape, and it looks really ugly and it’s really awful, but it’s there. So every single knot has to bound a disk in four-dimensional space.
And the question is, how nice is the disk? And this is really what comes about when you talk about sliceness. So in particular, you might ask the question: for a given knot, is it smoothly slice? And what that means is, does it bound a smooth disk? And a smooth disk means that it’s just really nice, no kinks in it anywhere, it’s very smooth, it has — all of the derivatives exist, which is the way you formally say that there aren’t any kinks in it. So that’s a question that people have asked about various knots over time. And in particular, there is this knot called the Conway knot that was due to John Conway when he was tabulating the knots, and I guess I didn’t say this before, but John Conway when he was in high school, he started tabulating the knots, and at that time, all they knew was the 10-crossing knots, and he figured out how to tabulate the 11-crossing knots.
(25:37) And so he was looking at the 11-crossing knots and he stumbled across this one knot that was really difficult to decide whether or not it was smoothly slice or not. And so he had this very difficult knot to deal with, but what made it difficult was that it was what’s called a mutant of another knot, called the Kinoshita-Terasaka knot. A very well-known knot, and he needed to figure out whether or not this particular knot, this new knot, that was a mutant of the other knot, was smoothly slice. And the Kinoshita-Terasaka knot is smoothly slice.
And interestingly enough, those two knots actually appear on the north gate and the south gate of the Mathematics Institute in Cambridge, because they were such famous knots, people were so intrigued by these two knots.
And so he put out this problem and he asked this question, and when he asked this question, this would have been in 1970 or so, ’74 maybe, and so, it’s, you know, a good 50 years ago that he asked this question, is this knot smoothly slice? Does this knot bound a disk in four-space that is this beautiful, smooth surface? And that question remained open for 50 years.
Strogatz (26:46): Wow. So, Colin, this has been absolutely wonderful. Thank you so much for spending time with us. I really feel like it taught us a lot.
Adams: Oh, thanks, Steve. It was really a lot of fun to talk to you.
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Strogatz (27:27): So, we’ve just heard a lot about knots and some of their uses from Colin Adams. Let’s talk now with someone who solved the 50-year-old problem about something called the Conway knot. My next guest, Lisa Piccirillo, did this on her own time as a graduate student, and she cracked it in about a week of intense work. Lisa Piccirillo is now an assistant professor of mathematics at MIT, where she specializes in the study of three-and four-dimensional spaces. Thank you so much for joining us today, Lisa.
Lisa Piccirillo (27:57): Thanks for having me.
Strogatz: I’m really excited about this. This is a great treat.
Your story got a lot of press over the years. It’s a sensational discovery that you made, and a really slick proof, beautiful argument. And I guess I’d love it if you could tell it as a story. Like, my understanding is you were at somebody’s — was it a birthday conference or something, when you first became aware of the question?
Piccirillo (28:21): Yeah, that’s right. I was at Bob Gompf’s birthday conference at UT [University of Texas] Austin, and it was like the end of the day on Saturday. I’d been to quite a few talks, and somebody was giving a talk on something a bit adjacent to what I do. But early on, you know, she was trying to convince people that her work is hard, and there’s lots of hard problems here. And she, she put up this slide. And she said, for example, we still don’t even know if this 11-crossing knot is slice.
And I know a bit about the field. But I don’t work in it too carefully. But, you know, there are tools, this is a mature field, I just thought, “Come on. Like, what do you mean, we don’t know that that 11-crossing knot is not slice?”
I guess I thought, “Well, it must not be that hard. It just must be that nobody cares.” So, but I thought, there are some people in my department who are interested in this sort of thing, and it was close to some work I was doing. So maybe I would try to use some tools I’d been developing to do this. And I’ll just show, like, the speaker, maybe. And she’ll say, wow, that’s nice. Good job. And that would be that.
Strogatz (29:31): So who was the speaker that you’re referring to?
Piccirillo: Oh, Shelly Harvey. She’s faculty at Rice.
Strogatz: Okay. And she’s a topologist, I assume.
Piccirillo (29:39): That’s right. And she really studies knot concordance as sort of her primary interest.
Strogatz (29:44): Huh. So Shelly gives this talk. You’re sitting there. At that point you’re how far into grad school?
Piccirillo (29:51): I was in my, the end of my fifth year. So I had one more year left, and I was going on the job market that fall.
Strogatz (29:56): Okay. And you mentioned that it was a Saturday. Is that the end of the conference?
Piccirillo (30:00): No, there were a couple of talks early in the morning on Sunday because the World Cup was on. And they were supposed to be at a normal time. But then people made them put the talks at like 7am so that they could go to the bar at like 10 to watch the World Cup.
Strogatz (30:15): This is it; the true secrets of mathematicians finally being revealed.
Piccirillo: Yeah, maybe I shouldn’t say that.
Strogatz (30:21): Okay, but so, you’re at this event, it’s a Saturday, you hear the talk. Like you — apparently you didn’t understand the magnitude of the question at the time.
Piccirillo (30:29): No, she just said it flippantly. Kind of like, yeah, we’re sort of bad at knot concordance because we can’t even do this.
Strogatz (30:35): Maybe this would be a good time for you to tell us a little, what the problem was, or why is Conway’s knot interesting?
Piccirillo (30:42): Yeah, so John Conway is this really legendary mathematician. He studied many things, knot theory, like, once or twice, but had a big impact on the field, in any case. And he built this knot, which is now called the Conway knot, in the late ‘70s. And he built it to be kind of a sneaky pathological knot. He was interested in some more three-dimensional properties of it, but it turned out he did a good job building a sneaky knot. So it’s hard, it’s kind of hard to deal with the Conway knot in almost all settings.
Strogatz (31:17): Can you tell us what’s sneaky about it?
Piccirillo (31:19): Yeah, what’s sneaky about it is that it has, like, a friend. Its friend is called the Kinoshita-Terasaka knot. There’ll be a quiz about that at the end. And it looks a lot like its friend, but they’re not actually the same knot. But it’s hard to tell that they’re not the same knot because they look so similar that the ways that we try to measure when knots are different, they can’t tell they’re different.
Strogatz (31:43): That’s very elegant. I love that. That’s a great summary, that this is the big issue in knot theory. I mean, Colin Adams mentioned that to us, when we were speaking to him, that one of the main enterprises is to figure out ways of distinguishing two knots, if they’re truly different. And you said that in the case of the Conway knot, the Conway knot was sort of artfully contrived to be ultimately very sneaky, a very good imposter to look a lot like this Kinoshita, what was it? Terasaka? What was it? I’m failing the quiz! Did I get that right?
Piccirillo (32:14): That’s exactly right.
Strogatz: So, that’s the point though. It’s like this artful imposter.
Piccirillo (32:17): That’s right. There’s a word for the type of imposter it is. It’s called a mutant of the Kinoshita-Terasaka knot.
Strogatz (32:23): Okay. So anyway, I mean, one thing that I’ve heard said about the Conway knot, is that, in terms of Conway’s contributions, one of the things he did was, people had understood or classified the knots up to 10 crossings. But all the knots with 11 crossings had not been classified apparently, or listed even, enumerated, until his work. And that he found this peculiar one with 11 crossings that could, as far as a certain invariant called the Alexander polynomial, or maybe even the Conway polynomial, that Conway’s knot was this fantastic imposter for a perfect circle, the Alexander polynomial couldn’t tell it apart.
Piccirillo (33:04): Yeah, that’s right. And even worse, maybe, the circle and the Kinoshita-Terasaka knot and the Conway knot all have the same Alexander polynomial.
Strogatz (33:14): Alright, so you hear about this problem from Shelley at the conference and you had this feeling… I forget exactly how you put it, but sort of like, “oh, come on, maybe I can solve this and show her.”
Piccirillo (33:28): Yeah, so I knew, right away, how I would try to show it. But there’s this kind of technical thing you have to do. So, I went to the conference dinner and had fun and didn’t do any more math. And then on Sunday, I think, maybe I didn’t go to watch the World Cup. And instead, I went to a café, and I started writing down the kind of technical thing I’d need to do, if I was going to show this knot wasn’t slice.
I don’t know, I worked for a few hours, and then put it away because I had, sort of, things I actually needed to be doing. And I just kind of kept following up on the evenings for the rest of the week, building this other knot. Maybe by midweek, I had the other knot. And then I computed something for the other knot. But I wasn’t doing this every night or staying up really late or anything.
Strogatz (34:11): All right, so maybe we should back up a little. And … I mean, because you’ve sketched what your strategy was, that you needed to compute a certain technical thing. And that would settle the question, and then you spent the rest of the week, I guess, computing it and everything worked, more or less. Which does make it sound easy, except that nobody else had been able to do it for 50 years.
Piccirillo (34:31): Well, the technical thing was not in vogue for 50 years. It’s probably still not in vogue, but it was what I was into. That’s maybe kind of where I had something that other people didn’t have, was that I study traces.
Strogatz (34:45): But we don’t yet know what a trace is. So maybe you could try to tell us a little, what is the trace of a knot?
Piccirillo (34:50): Sure. Something topologists do all the time is they use knots to build manifolds. A manifold is a space. In fact, for me, manifolds are the whole point of math, and knots are just a thing you use to get manifolds.
So that’s what I do with knots, I build manifolds out of them. And there’s a lot of ways to build a manifold out of a knot. If you think about a knot, maybe you’re thinking about it as being, kind of just floating in 3D space in front of you. And what you could do is you could look at its complement. So, the space around the knot, but not the knot itself. That’s this kind of interesting three-dimensional thing. So that’s a three-dimensional manifold associated to the knot.
A trace is a four-dimensional manifold associated to the knot, and — how should I tell you to think about the trace?
So let me do everything a dimension down first. If you had a knot embedded in two-dimensional space. So, you have a circle, maybe it looks kinda boring, that’s fine, you have a circle sitting on the table top. Then, you have some 3D space that sits below the table, all the air under there. And what you could do is you could take a bowl and put it upside down on top of the table, so that the rim of the bowl sits on your knot. And what we would call the 3D trace of that knot, which is sitting in the tabletop, would be the space below the table, plus the bowl.
You know, there’s the tabletop, that’s the 2D space that the knot lives in. There’s the air below the table. That’s 3D. And then there’s, like, this ceramic bowl, which is also 3D, sitting over the knot. So that would be the trace of a knot that was in 2D space.
Strogatz (36:16): That’s, it’s very illuminating to hear you trying to explain this. Admittedly, it’s very tough, and I want to congratulate you for doing it. But it’s, I think it’s revealing that you are using these analogies a dimension down, where we can picture things. Is that actually how you do this, when you’re working? Do you kind of go by analogy? Since I assume you can’t really picture these four-dimensional traces.
Piccirillo (36:53): That’s right. We call these analogies schematics, and all I do all day is draw schematics. And because — because of course, I can’t draw pictures of four-dimensional space. But I think that one of the things that drew me to topology, and four-dimensional topology in particular is — okay, topology is a really visual field, I really like that. And in three dimensions, sometimes we prove things by just, like, drawing the right picture of them.
And in four dimensions, that gets harder. Like, that ability to just literally observe the space goes away. But you can still draw these representative pictures where you lose some but not all information. And maybe if you draw many different pictures like that, if they each lose different types of information, then the set of pictures together, like, actually does still let you understand your space.
Maybe here’s something. Let me say what it means for a knot to be slice and then try to give you a little bit of intuition about why not all knots are slice. And I’ll use a very low-dimensional example to do it. A knot is slice, if it bounds a disk in four-dimensional space. So the disk gets to go kind of into that four-dimensional foam block, but the knot has to stay on its boundary, in three-space. And you can’t picture that, I can’t picture that, but that’s what it means.
So let’s try to picture what we maybe could picture which is, if we took, if we took our one-dimensional knot in the tabletop again, you know, you could find a disk that it bounds in the foam block below the table, because you could just sort of think about a bowl, you know, down there in the foam, and that would be a two-dimensional disk that the knot bounds.
Strogatz (38:29): Though actually, if I — to just pause for a second there, I don’t want you to lose your train of thought. But when you say disk, I think most people at this point are imagining a geometrical disk. So they would have thought of the flat part of the table underneath the bowl. But you — but in topology we don’t care about flatness. So you would say any bowl that has that circle as its boundary, including the disk that I’m picturing, but it could be a floppy thing that’s in the foam.
Piccirillo (38:54): Yeah, yeah, that’s right, I forgot to even make the distinction. All of my objects are made of rubber.
Strogatz (38:58): All right, but sorry, I interrupted you, so.
Piccirillo (38:59): So, and we were going to talk about zero-dimensional knots in one-dimensional space. So, a zero-dimensional knot is two points. And it lives in a line, if we’re letting it live in one-dimensional space. If a knot like that were going to be slice, what we would be asking for is a one-dimensional disk. So, like an interval, to have its boundary that knot.
I can’t build you a zero-dimensional knot that’s not slice, but I can build you a zero-dimensional link that’s not slice. So you’re going to think about your 1D space as being, like, the real numbers. And let’s take our first knot to be the points -1 and 1, and our second knot to be the points 0 and 2. So our knots are kind of intertwined.
Strogatz (39:43): Oh, that’s cool. So this is already the idea of what it means to be a link in one dimension. Is those two points, these pairs of points are interdigitated, where one of them is between the other two.
Piccirillo (39:53): Interdigitated is a really good word. Yes.
Piccirillo (39:58): So that one-dimensional space, we’re thinking of that as being, you know, the top of some piece of paper. And what we’d want to find is two intervals in our piece of paper that don’t intersect each other, and whose boundaries are two knots. Well, you can visualize trying to do that. And I think what you’re visualizing is it not going very well, right, those two intervals would have to hit each other if you want them to.
Strogatz (40:21): I’m picturing two chin straps.
Strogatz (40:24): Right, so there’s two points on a line, and they’re each one — each of the pairs of points has a chin strap hanging down, that goes from one point to the other point. And then when I interdigitate the other chin strap, the chin straps cross each other. And that’s bad? That’s no good?
Piccirillo (40:40): That’s bad. Yeah, you’re not allowed to have that. So I would call that link not slice because the chin straps are touching.
Strogatz (40:47): Oh, god, this is great. Thank you this, this was really helpful.
Piccirillo (40:51): Yeah, it’s sort of stunning that down there in dimension zero, you can learn something. But there we go.
Strogatz (40:56): Sounds like we need to talk more about the Rasmussen invariant, which you didn’t mention yet, and that kind of stuff.
Piccirillo (41:02): Sure. So the approach was, it was kind of in two parts. There’s this — and it started with this fact that I knew, because everybody who studies traces knows this fact, which is if you have a pair of knots that have the same trace, then they’re either both slice or they’re both not slice. So the thing that was immediate in the room, because I knew this fact, was that apparently the Conway knot is weird. So I don’t want to try to prove whether the Conway knot is slice or not, what I’m going to do is build another knot that has same trace as the Conway knot. And I’m going to study that other knot, and hopefully it won’t be so weird and hard to deal with.
So, the thing I went home to do was build the other knot. And that’s what took, I don’t know, until Thursday or something. But then, on Thursday night, I have this other knot in my hands. And now I just have to show that this other knot isn’t slice. And for that, well, I just used, as you mentioned, an invariant, one of these tools we use to detect properties of knots. There’s a kind of common invariant for obstructing sliceness called the “s” invariant, which was defined by Jake Rasmussen. And that’s what I used, then it worked.
Strogatz (42:07): And was it the first — like, you got lucky on the first try? Or did you try a few other invariants and they didn’t work?
Piccirillo (42:13): Well… both. I knew that a lot of other invariants couldn’t work. And there’s many invariants which, if the knots have the same trace, then the invariants match. So, since the Conway knot… In fact, I didn’t check, but since Shelly seemed to believe it was hard to show the Conway knot wasn’t slice, I assumed that she had checked the invariants for the Conway knot, so I shouldn’t use any that would sort of be preserved by the trace. And I had recently shown in some earlier work that the Rasmussen invariant didn’t have this property of being preserved by traces. So, it was a good choice.
Strogatz (42:48): Well, Lisa, thank you. This has been super fun, and mind-blowing, honestly. And all I can say is thank you very much for sharing. Really enjoyed talking with you.
Piccirillo (42:59): Thanks. It was a pleasure to be here.
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Strogatz (43:30): The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast, or in Quanta Magazine. The Joy of Why is produced by Susan Valot and Polly Stryker. Our editors are John Rennie and Thomas Lin. Our theme music was composed by Richie Johnson, and I’m your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected]. Thanks for listening.