OVER the course of one week in 2018, Lisa Piccirillo cracked a mathematical problem that had gone unsolved for half a century. Posed by legendary mathematician John Conway in 1970, it concerns a complex geometrical object known as the Conway knot. While an ordinary overhand knot â the kind you would tie at the end of a thread â sees the string cross over itself three times, the Conway knot has 11 crossings. What Conway wanted to know is whether his knot can be formed by cutting a slice out of a more complex four-dimensional knot â or, as mathematicians put it, is it âsliceâ?
Piccirillo discovered that it isnât. Her breakthrough came after finding a back door into the problem that could help mathematicians understand other four-dimensional objects. Currently a post-doctoral mathematician at Brandeis University in Waltham, Massachusetts, solving the Conway knot â along with her other research â has seen her offered a tenure-track position at the Massachusetts Institute of Technology. Âéśš´ŤĂ˝ spoke to her about the week she spent on the problem, her approach to mathematics and why it is time we stopped talking about geniuses.
Advertisement
Chelsea Whyte: How did you first become interested in mathematics?
Lisa Piccirillo: As a kid, I always liked maths and I was good at it in school. Iâm from quite a rural area in Maine, and people said âif you like maths, you can become an engineerâ. So I thought thatâs what you do with maths, become an engineer. I went to a lot of day camps for engineering and made a lot of bridges out of popsicle sticks, and found out that I didnât want to be an engineer. After that, I thought I didnât want to do maths.
But then I took calculus in college because I had to, and I had a professor that encouraged me to take the next class. By then, I had started getting hooked.
What was it that lit the spark?
Part of what got me hooked was learning that a field called topology, the study of shapes, existed. Itâs somehow a little more free-flowing, and I really liked the imaginative aspect of asking: âWhat can these shapes do?â It turns out that these complicated shapes can do some weird stuff, and understanding the realm of possibilities was a big draw.
How did you end up making maths your career?
The decision to go to graduate school was a difficult one. I still had this idea that I think a lot of people have, which is that the only way to be a successful mathematician is to be a genius, and Iâm certainly not anything like that. So I thought: âWhy bother? Iâm never going to be that good.â
Thereâs a strong stereotype of what people who do maths are like â introverted, nerdy, probably male, probably dead â and I was none of those things. I was very worried that I would have to give up other aspects of myself to be a maths robot and I didnât want to do that. I felt that tension very acutely in my undergraduate programme, but in graduate school, I learned that this tension isnât real. Mathematicians are interesting humans and none of them are geniuses.
Before I ask you about the Conway knot problem, can you tell me about knots more generally?
Let me back up and make a couple of definitions. I like to think about taking an extension cord out of the basement where itâs been for a while. Itâs probably a hot mess, and if we just plug the ends together, it will still be a hot mess. We say a knot is âtrivialâ if it isnât a hot mess â that is to say, if itâs possible to untangle it without unplugging the ends. In classical knot theory, trivial means you could move it around to look like the rim of a dinner plate. Mathematicians like to say âit bounds a discâ. What makes things confusing is that while the knot has to live in three dimensions, the disc it bounds doesnât have to; it could live in four dimensions, for example.
When you say four dimensions, I think of the three dimensions of space, plus time. Is that too literal?
Yes, thatâs too literal. We just have four independent directions to work with. It doesnât really matter to us what they correspond to in the real world.
When you are solving these problems, do you picture four dimensions in your head?
Iâm only ever thinking about 3D spaces because thatâs all I can visualise, just like everybody else. Let me give you an example. Imagine that you and I existed in a 2D universe â so a flat plane like a sheet of paper â and a hollow beach ball came through our world. If only the very bottom of the ball made contact with our world, from our perspective, it would look like a point.
But if more of the ball entered our 2D universe, it would look like a circle. As it kept passing through, we would see larger and larger circles, and then smaller circles, and then a point at the top of the ball and then nothing. You have point, circles, point again. Of course, in 3D space, it is far easier to just picture an entire beach ball instead of cutting it up into 2D slices. But when I want to think about an object in 4D space, I canât just picture it, so we use this trick of cutting it up into 3D slices.
Do you slice things up in the same way when you think about knots?
Yes. If a knot in 3D space bounds a disc in 4D space, we say it is âsliceâ. The knot lives where we do, in three dimensions, but the disc is allowed to use another space. So the Conway knot problem is just: does this particular knot with 11 crossings bound a disc in four dimensions? Is it âsliceâ? Itâs yes or no.
How did you first hear about Conwayâs knot?
It was in a talk at a conference at the end of July 2018. The speaker mentioned that it was still an open problem. I thought that was ridiculous: itâs 2018, we know a lot about sliceness, whether this 11-crossing knot is slice shouldnât be an open question. The ridiculousness of the problem is what made me think about it.
But I really didnât know very much about it. I thought it was just quite esoteric. Itâs an open problem, but I thought, âprobably the reason itâs open is mostly because nobodyâs tried very hardâ.
Now that you have solved it, do you still think that?
No, apparently thatâs not true. It took a specific tool that I happened to have been developing.
One way that mathematicians describe four-dimensional shapes â what we call 4-manifolds â is by making knots in 3D space and using the knots as sort of instructions for how to build them.
All knots have something called a trace, which is the manifold you can build from that knot. I knew that if you have two knots with the same trace, theyâre either both slice or both not slice. That has been known by mathematicians for a long time. This fact was very present in my mind, because I use it for my study of knot traces.
I knew that if I could build a second knot that shared a trace with the Conway knot and that happened to be slice, then Iâd have solved the problem. While thatâs a technical thing to do, it just takes a bit of calculation.
âYou do maths because you love it on the days when you donât prove anythingâ
How long did it take you?
I learned about the problem on a Saturday and I certainly knew the answer by the next Saturday. And I thought: âNobodyâs going to care about this.â I was only working in the evenings. I wasnât tearing my hair out and burning the midnight oil.
My idea worked right away. I guess it was overlooked because people werenât really studying traces and this calculation isnât completely trivial â it uses tools that Iâd developed in other work. But I think anyone who had my technical knowledge could have solved it quickly too.
Did you know how big a deal this was when you had solved it?
No, I thought it would go in a very low-tier journal, or perhaps I wouldnât try to publish it at all.
Did you go into the field of maths to solve big problems like this?
In maths, 100 per cent of the days, basically you wonât solve anything. So you have to learn to be okay with that and still enjoy what youâre doing, even though today you wonât answer anything, and tomorrow you also wonât answer anything and the same thing will be true for the rest of your life except for a few good days. You have to be doing maths because you love it on the days when you didnât prove anything. The good days are so far apart. It doesnât matter how good they are. If that were the reason I was in it, I know I wouldnât make it.
That reminds me of the conceptual artist John Baldessari, whose advice to young artists was: âYou have to be possessed, which you canât will.â
Yes. Itâs more fun when youâre possessed too.
Forgive me for asking this, but why does any of this matter?
The reason a lot of mathematicians â myself included â care about sliceness is because it helps us understand 4D spaces. One of the major challenges in 4D topology is distinguishing between simple 4D spaces. Generally, this is pretty hard because there arenât many tools available. Traces provide a tool.
If you have two different 4D spaces, and both of them have some 3D space on the boundary, itâs very possible that a knot in 3D space can bound to a disc in one 4D space but not the other. That can help us understand differences between two 4D spaces in a way that would otherwise be very complicated.
What will you be working on next?
Iâm still very interested in 4-manifolds and in using sliceness to understand them better. Itâs also true that this trick I used for the Conway knot doesnât work on some other, more complicated knots. The reason is because it isnât always possible to build a trace â sometimes itâs provably impossible or we just donât know how to do it.
Iâm trying to understand how to apply this type of argument more broadly to sliceness problems. More concretely, it turns out that sometimes, for some special knots, I can go home and build you another knot that shows a trace, but a computer canât. Why not? Itâs because we donât know the rules of how we do it ourselves. If the maths gods hand me a knot and ask me to build a trace, I may get lucky, but I donât know if I could tell you how I got there. And Iâd like to understand why.
