Solving a decades-old mathematical mystery
When Lisa Piccirillo was still a student, she solved a 50 year old math mystery in a single week. When you hear that, you might imagine Dr. Piccirillo as a math prodigy. But while Dr. Piccirillo definitely considers herself to be someone who loves math, she doesn’t fit this stereotype. As a creative woman who loves drawing and woodworking, she isn’t what most people picture when they imagine a mathematician. She didn’t even imagine herself as one until she was in college. But after a math class with an encouraging professor and a fascinating talk during her first year, she fell in love with the subject, and is now a mathematics professor at MIT.
Lisa Piccirillo’s research focuses on a type of math called topology, which sometimes includes studying knots. These knots aren’t just in rope or in long hair – they can be complex objects that mathematicians like Dr. Piccirillo work to untangle!
What is topology?
How do we know when two objects have the same shape? It’s pretty easy to tell that a ball, a donut, and a coffee mug have different shapes, but what about a donut and a hula hoop? They may have different sizes, but they look pretty similar – you can imagine stretching the donut to turn it into the hula hoop.
In topology, mathematicians try to get a concrete answer. They want to know which objects are the same under what they call continuous deformations, which are specific ways to reshape objects. Stretching, shrinking, twisting, and bending are allowed, but cutting, gluing, and tearing are not. Mathematicians want to understand what properties of objects stay the same even when the objects are squished like play-doh.
Dr. Piccirillo studies topology in three dimensions, and also in four dimensions! Working in three dimensions is just working in the space around us – you can move forward and backward, left and right, up and down. Thinking about shapes in four dimensions means adding another direction to this. While you can’t really visualize this in space, you can try to imagine it by thinking about a fourth “time” dimension by studying how three dimensional shapes change in time.
How to define a knot
One type of complex shape that Lisa Piccirillo studies is called a knot. Imagine taking an extension cord or a piece of string, tangling it around itself, and sticking the ends together. Anything you can make this way is a knot. The simplest example is just a loop of string with no tangles. This is called the unknot. This may not seem like what we usually think of when we think of a knot, but in knot theory it’s the mathematical equivalent of the number zero. There are also lots of examples of more complicated knots which can have many tangles. The ways in which these knots in 3D space can be drawn on 2D paper are called knot projections or knot diagrams.
Just like donuts and hula hoops have the same shape, some knots that look different are actually the same knot. For example, you can imagine taking the unknot (which remember is just a simple loop) and twisting a piece of it. Since it’s easy to get back and forth between this knot and the unknot, we can think of them as the same knot. More precisely, we say that two knots are the same if we can continuously deform one into the other without cutting or gluing.
The next natural question to ask is: how can we tell if two knots are the same? If we have very simple knots, maybe we can play around with them for a while and turn one into the other. But when we have more complicated knots, or when two knots aren't the same, how do we use math to prove that?
A common way topologists and knot theorists do this is by looking at the invariant properties of the knots. These are things about the knot that don’t change, no matter how we represent that knot in 2D or 3D space. This means invariant properties are not changed by continuous deformations like twisting and pulling. So if two knots have a different value of an invariant, they must be different knots. But if two knots have the same value of an invariant, that isn’t enough to say that they’re the same! It’s possible that they’re the same in some ways, but there’s another property that’s different which we just haven’t found yet.
One example of an invariant is called the crossing number. The crossing number is what it sounds like – it’s the number of times the string crosses itself! More precisely, for a drawing of a knot on a piece of paper, we can ask how many times the string vanishes underneath itself, so that we can’t see it. And while this number might depend on how you smoosh or twist the knot, the crossing number is defined as the minimum number of crossings that knot has for any way we can draw it.
Another example of an invariant is tricolorability. This invariant classifies knots by coloring them with multiple colors, with the color only changing at crossings. More precisely, if you can color a knot with at least two colors such that every crossing has either a single color or three different colors coming out of it and the color only changes when you can’t see the string (when it vanishes under another piece of the string on the page), then that knot is tricolorable!
What does it mean for a knot to be slice?
A more complicated invariant property of knots is whether or not they are slice. This is a property related to whether a knot can become the boundary of a disc. Imagine that you have a round tablecloth and some knot in a piece of string, but you want the string to run along the edge of the table cloth. Depending on how tangled the knot is, it may be possible to twist the tablecloth so that you can sew the string onto the edge of the circular cloth. If it is, then we say that the knot bounds a 2D disk (the round tablecloth) in 3D (since you have to twist the cloth in a third dimension to follow the knot). One example of a knot that does this is the unknot, since it’s just a simple loop that can go along the outside of a tablecloth without any twisting.
To say that a knot is slice is similar to this, except we imagine the two dimensional table cloth has some extra space to twist around, and let that extra space be an extra dimension. So the mathematical definition of a slice knot is a knot that can bound a 2D disk (our round tablecloth) if we let the disk move around in 4 dimensions.
This is a challenge to visualize since humans aren’t wired to think in four spatial dimensions. To help us understand, we can imagine time as the fourth dimension, and study how a 3D surface changes over time. In this picture, we can imagine filming a movie as we manipulate our knot, and the knot at any given time in that video corresponds to a cross section of a higher dimensional surface. For example, if we imagine we had two dimensional shapes instead of knots, we could film a movie turning a circle into a triangle, which would correspond to a tube with a triangle on one end and a circle on the other.
We can make a similar video with 3D knots changing in time, with one knot at the top and a different knot at the bottom. Unlike the 2D case, we can’t completely visualize the surface here, but we can use the color of the surface to represent the height of the knot above the paper it’s drawn on. We can ask more generally about what transformations we can make by moving a knot along a 4D surface. It turns out that even if we require the surface to be smooth (not have any holes or tears), there are still operations that are allowed in four dimensions that weren’t allowed for regular knots, such as cutting or gluing the cord, or making a loop vanish.
One question we can ask is whether a given knot can be “untangled” into the unknot along our 4D surface. It turns out that this is the same question we asked before when talking about the table cloth: if it’s possible to turn a knot into the unknot through a surface in 4D, then that knot is slice. While you can’t tell just from looking at it, the knot shown to the right is an example of a slice knot.
The Conway knot problem
With all these tools to categorize knots, mathematicians like Dr. Piccirillio can tackle precise problems, like seeing if a particular knot has a specific property. But it’s not always easy! Sometimes, these kinds of questions can stump mathematicians for decades before someone comes up with the right way of looking at the problem.
One such mathematical mystery was called the Conway knot problem. It asked whether a particular knot, called the Conway knot, is slice. For 50 years, mathematicians couldn’t figure out the answer. Part of why this was so hard is because the Conway knot is very similar to another knot that was known to be slice, and it was challenging to find invariant properties that were different between the two knots that would mean one was slice but the other wasn’t.
When Lisa Piccirllio was a graduate student, she showed that the Conway knot was in fact not slice, using a creative technique that hadn’t been applied to this problem before. It turns out that knots can be associated with a particular 4D space, and if the 4D spaces of two knots match, then they must have the same sliceness. Taking advantage of this property, Dr. Piccirillio came up with a third knot that shared the same 4D space as the Conway knot. Since the 4D spaces matched, the sliceness of this new knot and the mysterious Conway knot had to be the same, and since the new knot was not slice, neither was the Conway knot.
Since solving the Conway knot problem, Dr. Piccirillio has continued to work on other hard problems in topology. Her passion for the subject drives her to tackle challenges, even when she sometimes feels stuck. Mathematical research sometimes makes fast progress, but many problems wait unsolved for years and years. But Dr. Piccirillio’s story shows that a little creativity can lead to exciting and unexpected progress!
Written by Katherine Fraser
Edited by Caroline Martin
Illustrations by Lindsey Oberhelman
A Tough Knot to Crack by Boston College Magazine
Graduate Student Solves Decades-Old Conway Knot Problem by Quanta Magazine
Notes on Tricolorability by Xiaoyu Qiao
A 50 year-old enigma solved: the Conway knot is not slice by Mickaël Launay (French with English Subtitles)
How You Too Can Solve 50+ Year Old Problems by Talks at Google
If you’d like to learn more about the Conway knot problem and Dr. Piccirillo’s work, watch the two videos linked above. To find out about knot theory and topology more generally, try these books:
The Shape of Space by Jeffrey R. Weeks
The Knot Book by Collin C. Adams
Learn more about knots and topology!
Investigate (30-90 mins): Grab three different color pencils and figure out whether or not these knots are tricolorable.
Build (20-40 mins): Find some string or rope and try tangling and untangling these knots for yourself.
Explore (15 mins): Try this trick with coins to learn more about how shapes behave in different numbers of dimensions.