Graduate student solved the problem of “Conway Node”, over which they fought for decades

It took Lisa Piccirillo less than a week to answer the long-standing question about a strange site discovered more than half a century ago by the legendary John Conway.

In the summer of 2018, at a conference on low-dimensional topology and geometry, Lisa Picchirillo heard about a cute little mathematical problem. It looked like a good testing ground for some of the methods she developed as a graduate student at the University of Texas at Austin.

“I did not allow myself to work on this day,” she said, “because I did not consider this to be real mathematics. I thought it was my homework. ”

The question is whether the Conway node - a lever opened more than half a century ago by the legendary mathematician John Horton Conway - is a piece of a higher-dimensional node. “Sliceness” is one of the first natural questions that knot theorists ask about knots in spaces with higher dimensions, and mathematicians were able to answer it for all thousands of knots with 12 or less intersections, with the exception of one. The Conway node, which has 11 intersections, has been teasing mathematicians for decades.



The solution to the problem of the Conway node, proposed by Lisa Pichchirillo, helped her get a permanent position at the Massachusetts Institute of Technology.


In less than a week, Picchirillo already had the answer: the Conway node is not a “slice”. A few days later, she met with Cameron Gordon, a professor at the University of Austin, and in passing mentioned her decision.

"What?! It’s getting into the annals right now! ” - Gordon said, referring to the "Annals of Mathematics", one of the best journals in this discipline.

He began to shout: “Why aren’t you jumping for joy?” Says Picchirillo, now a graduate student at Brandeis University. “He even scared a little.”

“I don’t think she understood what an old and famous problem it is,” Gordon said.

Piccirillo Proofappeared in the Annals of Mathematics in February. This article, combined with her other work, provided her with a permanent job offer at the Massachusetts Institute of Technology, which begins July 1, just 14 months after she completes her doctorate.

The question of the cut-off of the Conway knot was known not only because of how long it remained unresolved. Slices of knots give mathematicians the opportunity to explore the strange nature of four-dimensional space in which two-dimensional spheres can be tied in a knot, sometimes in such a crumpled way that they cannot be smoothed out. According to Charles Livingston, professor emeritus at Indiana University, sheerness "is right now linked to some of the deepest issues in four-dimensional topology."

“This question of whether Conway’s knot is a slice was a kind of breakdown for many modern developments in the general field of knot theory,” said Joshua Greene of Boston College, who oversaw Picchirillo’s graduation thesis during her student years. “I was very pleased to see how someone I had known for so long suddenly pulled a sword from a stone.”

Magic ball

While most of us think that a knot exists in a piece of string with two ends, mathematicians think that these two ends are connected, so the knot cannot be untangled. Over the past century, these nodal cycles have helped illuminate various topics ∞ from quantum physics to the structure of DNA, as well as the topology of three-dimensional space.


John Conway explained in 1990 how he showed in high school why two nodes cannot balance each other.

But our world is four-dimensional if we include time as a measurement, so it’s natural to ask if there is a corresponding theory of knots in 4D space. It’s not just about taking all the nodes that we have in three-dimensional space and immersing them in 4D space: with four dimensions to move in a circle, any knotty loop can be unraveled if the threads move one above the other in the fourth dimension .

To make a knotty object in four-dimensional space, you need a two-dimensional sphere, not a one-dimensional loop. Just as the three dimensions provide enough space for building knotted loops, but not enough room for them to unravel, the four dimensions provide such an environment for the knotted spheres that mathematicians first built in the 1920s.

It's hard to visualize a knotted sphere in 4D space, but it helps to think about a regular sphere in 3D space first. If you cut through it, you will see an loose loop. But when you cut a knotted sphere in 4D space, you can see a knotted loop (or, possibly, an unrecognizable loop or a link of several loops, instead, depending on where you cut). Any node that you can make by cutting a knotted sphere is called a “slice”. Some nodes are not cut, for example, a three-junction node, known as a trefoil.

Cut nodes “provide a bridge between three-dimensional and four-dimensional histories of node theory,” Green said.

But there is a wrinkle that gives the richness and originality of a four-dimensional story: in the 4D topology there are two different versions of what it means to be cut. In a series of revolutionary developments in the early 1980s (which brought medals to Michael Freedman and Simon Donaldson Fields), mathematicians found that the 4D space contains not only smooth spheres that we intuitively visualize, but also spheres that are so pervasively crumpled that they could never be ironed smoothly. The question of which nodes are a slice depends on whether you decide to include these crumpled spheres.

“These are very, very strange objects that seem to exist through magic,” said Shelly Harvey of Rice University. (It was at Harvey’s speech in 2018 that Picchirillo first learned about the problem of the Conway node.)

These strange areas are not a mistake of the four-dimensional topology, but a feature. Nodes that are “topologically cut” but not “smoothly cut” —that is, they are a cut of some crumpled sphere, but not smooth — allow mathematicians to construct so-called “exotic” versions of ordinary four-dimensional space. These copies of four-dimensional space look the same as normal space from a topological point of view, but are irretrievably crumpled. The existence of these exotic spaces distinguishes the fourth dimension from all other dimensions.

The issue of smoothness is the “lowest dimensional sensor” of these exotic four-dimensional spaces, Green said.

Over the years, mathematicians have discovered a number of nodes that were topologically, but not smoothly cut. However, among the nodes with 12 or fewer intersections, there seemed to be none — with the possible exception of the Conway node. Mathematicians could calculate the cutoff state of all other nodes with 12 or less intersections, but the Conway node eluded them.

Conway, who died of COVID-19 last month, was known for making influential contributions to one area of ​​mathematics after another. He first became interested in nodes in his teens in the 1950s and came up with a simple way to list almost all nodes up to 11 intersections (previous complete lists only reached 10 intersections).

There was one node on the list that stood out. “Conway, I think, understood that there was something completely special about this,” Green said.

The Conway node, as they began to call it, is topologically cut off - mathematicians understood this against the backdrop of the revolutionary discoveries of the 1980s, but they could not understand whether it was smoothly cut. They suspected that this was not so, because he seemed to lack a function called "ribbing," which smoothly cut knots usually have. But he also had a feature that made him immune to any attempt to show that he was not smoothly cut.

Namely, the Conway node has a kind of relative - the so-called mutant. If you draw a Conway knot on paper, cut out a specific piece of paper, flip a fragment over and then connect its free ends, you will get another knot, known as the Kinoshita-Terasaka knot .

The trouble is that this new unit turned out to be smoothly cut. And since the Conway node is so closely connected with the smooth cut node, it manages to deceive all the tools (called invariants) that mathematicians use to find nodes without a cut.

“Whenever a new invariant appears, we try to check it on the Conway node,” Green said. “This is just one stubborn example, which, regardless of which invariant you come up with, does not tell you whether this is a slice or not. ".

Conway's node “is located at the intersection of the blind spots” of these various instruments, Piccirillo said.

One mathematician, Mark Hughes of Brigham Young University, created a neural network that uses node invariants and other information to predict characteristics such as shearness. For most nodes, the network makes clear forecasts. But what is his hunch about whether Conway's knot is smoothly cut? Fifty fifty.

“Over time, it turned into a knot that we could not cope with,” Livingston said.

Smart ups and downs

Piccirillo enjoys the visual intuition that knot theory entails, but she does not think of herself primarily as a knot theorist. “These are really [three-dimensional and four-dimensional forms] that excite me, but the study of these things is deeply connected with the theory of knots, so I also do a little of this,” she wrote in an email.

“When she first started studying math in college, she didn't stand out as a“ standard golden child math prodigy, ”said Elisenda Grigsby, one of Pichchirillo’s professors at Boston College. Most likely, it was creativity Pichchirillo attracted the attention of Grigsby. "She really believed in her point of view, and that has always been so."

Piccirillo was confronted with the question of the Conway node at a time when she was thinking about a different way of connecting two nodes besides a mutation. Each node has a corresponding four-dimensional shape, called its trace, which is created by placing the node on the border of the 4D ball and sewing on it a kind of cap along the node. The node trace “encodes this node very strongly,” Gordon said.



One of the former professors, Piccirillo called creativity - one of its main strengths as mathematics.

Different nodes can have the same four-dimensional trace, and mathematicians already knew that these twin brothers of the trace, so to speak, always have the same slice status - either they are both slice or both are not slice. But Piccirillo and Allison Miller, now Rice's graduate student, have shown that these trace siblings do not necessarily look the same for all knot invariants used to study smoothness.

This pointed to Picchirillo's strategy of proving that the Conway node is not a slice: if it could build trace affinity for the Conway node, it would probably work better with one of the cut invariants than the Conway node.

Creating trace brothers and sisters is a complex matter, but Picchirillo was a true expert. “This is just my profession,” she said. “So I just went home and did it.”

Thanks to a combination of ingenious turns, Piccirillo managed to build a complex knot that has the same footprint as the Conway knot. For this node, a tool called the Rasmussen s-invariant shows that it is not a smooth cut - so the Conway node cannot be either one or the other.

“This is really great evidence,” Gordon said. According to him, there was no reason to expect that the node constructed by Picchirillo would yield to Rasmussen's s-invariant. "But it worked ... somehow surprisingly."

Piccirillo’s evidence “fits into the form of short, amazing evidence of elusive results that researchers in this field are able to quickly absorb, admire, and strive to generalize - not to mention wondering how it took so much time,” Green wrote in an email .

“Footprints are a classic tool that has been around for decades, but Picchirillo understood more deeply than anyone else,” admires Green. “Her work showed topologists that footprints are underestimated. She picked up some tools on which, perhaps, a little dusty. Others are now following his example. ”

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