Get best of the week

Graduate student solved a topological problem half a century ago

It took Lisa Picchirillo less than a week to find an answer to the old question about a strange node discovered more than fifty years ago by the legendary mathematician John Conway.




In the summer of 2018, at a conference on low-dimensional topology and geometry, Lisa Picchirillo heard about a small mathematical problem. She seemed like a good testing ground for some of the techniques that Lisa developed as a graduate student at the University of Texas at Austin.

“I did not allow myself to work on it during the day,” she said, “because I did not consider this task to be real mathematics. I perceived her more as homework. ”

The question was: is the Conway knot - a complex weave of rope, discovered more than fifty years ago by the legendary mathematician John Horton Conway - a slice of a higher dimensional knot. “Sliceness” is one of the first natural questions that specialists inknot theories ask about knots from high-resolution spaces, and mathematicians were able to answer it for many thousands of knots with no more than 12 intersections - all but one. The Conway node, which has 11 intersections, has teased mathematicians for many decades.

Before the end of the week, Picchirillo was ready to answer: the Conway node is not the mentioned section. A few days later, when meeting with Cameron Gordon, a professor at the University of Texas, she casually mentioned her decision.

"I said that?? Yes, it should immediately go to the Annals! ” - said Gordon, referring to one of the largest mathematical journals, Annals of Mathematics.

"He began to shout: Why are you not happy about this?" Said Pichchirillo, now a postdoc at Brandeis University. "He's like crazy."

“I don’t think she realized how old and famous this task was,” Gordon said.

EvidencePiccirillo appeared in the journal Annals of Mathematics in February. This work and her other achievements provided her with a place at the Massachusetts Institute of Technology, where she will start working on July 1, only 14 months after she defended her doctorate.

The question of whether the Conway node belonged to the cut-off was famous not only because it remained unanswered for so long. Cut off knots give mathematicians the opportunity to probe the strange nature of four-dimensional space, in which it is sometimes possible to knit two-dimensional spheres into a knot in such a crumpled way that they cannot be smoothed out. Sobriety is “related to some of the deepest issues of four-dimensional topology,” said Charles Livingston , professor emeritus at Indiana University.

“The issue of conway knot shearness has been a criterion for many modern developments related to general aspects of knot theory,” said Joshua Green of Boston College, graduate supervisor Piccirillo. “It was very nice to see how a man whom I had known for quite some time suddenly pulled this sword from a stone.”

Magic spheres


Most of us imagine the knot as a piece of intertwined rope with two ends. However, mathematicians work with ropes whose ends are interconnected, as a result of which the knot cannot be untangled. Over the past century, these knotted loops have helped to study questions from various fields of science, from quantum physics to the structure of DNA, as well as from the topology of three-dimensional spaces.


In this 1990 video, John Conway explains how, in high school, he showed that two nodes do not cancel each other out.

However, if we take time into account as a measurement, our world will be four-dimensional, so it’s natural to ask about the existence of an appropriate theory of nodes in 4D. And this does not mean that we can simply take all three-dimensional knots and shove them into four-dimensional space: if you have four dimensions, you can unravel any loop if you start lifting the pieces of rope above each other in the fourth dimension.

To tie a knot in 4D, you need a two-dimensional sphere, not a one-dimensional loop. Just like the three dimensions provide enough room for tying loops, but not for untying them, the four dimensions provide a place for tying spheres, which mathematicians first did in the 1920s.

It is difficult to imagine a tied sphere in four-dimensional space, but for this it is useful to first imagine a normal sphere in 3D. If you cut it, you will see an unbound loop. But if you cut the connected sphere in 4D, you will see a connected loop (or, possibly, an unconnected loop, or several loops connected to each other - it depends on where to cut). Any node that can be obtained by cutting a connected sphere is considered cut off. Some nodes are not cut off - for example, a node with three intersections, trifolium.



Cut nodes “bridge the three-dimensional and four-dimensional stories of knot theory,” Green said.

However, there is one problem that reveals the richness and specificity of four-dimensional history: in four-dimensional topology, there are two different options for shearness. Several revolutionary works in the early 1980s (for which Michael Friedman and Simon Donaldson received the Fields Prize) showed that four-dimensional space contains not only smooth spheres that we intuitively imagine. It also has crumpled spheres that cannot be smoothed out. And the question of the knot cut-off depends on whether to consider these crumpled spheres.

“These are very, very strange objects, almost magical,” said Shelley Harveyfrom Rice University (it was from Harvey's report in 2018 that Piccirillo first learned about Conway's node).

These strange spheres are not an error of the four-dimensional topology, but its peculiarity. Topological cut-off, but not “smoothly cut” knots - that is, knots that are slices of crumpled spheres - allow mathematicians to create so-called “Exotic” variants of the usual four-dimensional space. From a topological point of view, these copies of four-dimensional space look the same as usual, but at the same time they are irrevocably crumpled. The existence of such exotic spaces distinguishes the fourth dimension from all others.

The cut-off issue is the “smallest probe” for these exotic four-dimensional spaces, Green said.

Over the years of research, mathematicians have discovered a whole set of nodes cut topologically, but not smoothly. However, among nodes with the number of intersections of up to 12 such, it seems, was not observed - with the possible exception of the Conway node. Mathematicians could figure out the cutoff of all the other nodes with the number of intersections not higher than 12, however, they were not given the Conway node in any way.

Conway, who died last month due to a coronavirus, was known for his important contributions to a wide range of areas of mathematics. He became interested in nodes for the first time in the 1950s and came up with a simple way to list almost all nodes with the number of intersections up to 11 (previous complete lists included only nodes with the number of intersections up to 10).

But one node on this list stood apart. “I think Conway realized that this node was somehow special,” Green said.

The Conway node, as it was later called, is a topological section - mathematicians understood this back in the 1980s as part of a series of revolutionary discoveries. However, they could not figure out whether it was a smooth cut. They suspected that this was not so, since he did not have such a feature as the “ribbon” that is usually observed in smooth nodes. However, another feature of it did not give a chance to all attempts to show that this cut is not smooth.

Namely, the Conway node has a fraternal node, or, as they say in the theory of nodes, a mutation. If you draw a Conway knot on paper, cut out a certain part of it, flip a fragment and reconnect the knot, you get another knot, known as the Kinoshita – Terasaki knot .


To prove that the Conway node is not a smooth cut, scientists were prevented by its similarity to the Kinoshita – Terasaki node. Lisa Picchirillo figured out how to tie a new, more complex companion to the Conway node.

The problem is that this new node is a smooth cut. And since the Conway node is so much like a smooth slice, it avoids the effects of all the tools (invariants) used by mathematicians to determine nodes that are not slices.

“When a new invariant appeared, we tried to test it on the Conway node,” Green said. “And this is such a unique stubborn example, which, regardless of the invariant, did not tell us whether it is a slice or not.”

Conway’s knot “falls into the intersection of blind spots” of these instruments, Picchirillo said.

One mathematician, Mark Hughes of Brigham Young University, created a neural network using node invariants and other information to predict properties such as shearness. For most nodes, the network makes clear predictions. Do you know what she said about the smooth cut-off of the Conway knot? 50 to 50.

“Over time, this knot began to stand out among others as not subject to us,” Livingston said.

Tricky Turns


Picchirillo likes the visual intuition associated with knot theory, but she does not think that she is primarily a theorist in this field. “I’m more interested in three-dimensional and four-dimensional figures, but their study is closely intertwined with the theory of nodes, so I’m doing it a bit,” she wrote in an email.

When she began to study mathematics in college, she did not stand out as “a standard child prodigy in mathematics,” said Elisenda Grisby , one of Picchirillo ’s teachers at Boston College. Grisby first noticed the creative nature of Picchirillo. "She always believed in the correctness of her point of view."

The question related to Conway's node came to Pichchirillo when she thought about whether the nodes could be connected by anything other than mutations. Each node has its so-called. a four-dimensional trace that can be obtained if you place the knot on the border of the four-dimensional ball and sew something like a hood on top along the knot. The footprint of the node “encodes its node fairly hard,” Gordon said.



Different nodes may have the same four-dimensional trace, and mathematicians already knew that such, so to speak, relatives in the tracks always have the same cut status - either they are cut or not. However, Piccirillo and Allison Miller , a postdoc from Rice University, showedthat such trace relatives do not necessarily look the same for all invariants used to study shearness.

This indicated Picchirillo the path to the strategy used to prove that the Conway node is not cut off: if she could create a trace relative for this node, perhaps he would be more willing to cooperate with one of the cut invariants than the Conway node itself.

The construction of such relatives is a difficult task, but Picchirillo was an expert in this. “I'm basically doing this,” she said. “So I just went home and did it.”

Using an ingenious combination, Pichchirillo was able to construct a complex node that has the same trace as the Conway node. And for this node, a tool called
Rasmussen's “c-invariant” shows that it is not smoothly cut off - as, therefore, is the Conway node.

“Very beautiful proof,” Gordon said. According to him, there was no reason to expect that the node created by Picchirillo would succumb to the Rasmussen c-invariant. “However, the approach worked, which is even surprising.”

Pichchirillo’s evidence “goes along with short and unexpected evidence of elusive results that researchers in this field can quickly digest, admire and try to generalize - not to mention wondering why no one could think of this for so long,” wrote by Green in the email.

Footprints are a classic tool that has existed for several decades, but Picchirillo figured it out better than others, Green said. According to him, her work showed topologists that the traces of nodes are underestimated. “She took some slightly dusty tools,” he said. “And now others are already following her example.”

More articles:


All Articles