A Simple Way To Measure Knots Has Come Unraveled
Samuel Velasco/Quanta Magazine
Introduction
In 1876, Peter Guthrie Tait set out to measure what he called the “beknottedness” of knots.
The Scottish mathematician, whose research laid the foundation for modern knot theory, was trying to find a way to tell knots apart — a notoriously difficult task. In math, a knot is a tangled piece of string with its ends glued together. Two knots are the same if you can twist and stretch one into the other without cutting the string. But it’s hard to tell if this is possible based solely on what the knots look like. A knot that seems really complicated and tangled, for instance, might actually be equivalent to a simple loop.
Tait had an idea for how to determine if two knots are different. First, lay a knot flat on a table and find a spot where the string crosses over itself. Cut the string, swap the positions of the strands, and glue everything back together. This is called a crossing change. If you do this enough times, you’ll be left with an unknotted circle. Tait’s beknottedness is the minimum number of crossing changes that this process requires. Today, it’s known as a knot’s “unknotting number.”
If two knots have different unknotting numbers, then they must be different. But Tait found that his unknotting numbers generated more questions than they answered.
“I have got so thoroughly on one groove,” he wrote in a letter to a friend, the scientist James Clerk Maxwell, “that I fear I may be missing or unduly exalting something which will appear excessively simple to anyone but myself.”
Mark Belan/Quanta Magazine
If Tait missed something, so did every mathematician who followed him. Over the past 150 years, many knot theorists have been baffled by the unknotting number. They know it can provide a powerful description of a knot. “It’s the most fundamental [measure] of all, arguably,” said Susan Hermiller of the University of Nebraska. But it’s often impossibly hard to compute a knot’s unknotting number, and it’s not always clear how that number corresponds to the knot’s complexity.
To untangle this mystery, mathematicians in the early 20th century devised a straightforward conjecture about how the unknotting number changes when you combine knots. If they could prove it, they would have a way to compute the unknotting number for any knot — giving mathematicians a simple, concrete way to measure knot complexity.
Researchers searched for nearly a century, finding little evidence either for or against the conjecture.
Then, in a paper posted in June, Hermiller and her longtime collaborator Mark Brittenham uncovered a pair of knots that, when combined, form a knot that is easier to untie than the conjecture predicts. In doing so, they disproved the conjecture — and used their counterexample to find infinitely many other pairs of knots that also disprove it.
“When the paper was posted, I gasped out loud,” said Allison Moore of Virginia Commonwealth University.
The result demonstrates that “the unknotting number is chaotic and unpredictable and really exciting to study,” she added. The paper is “like waving a flag that says, we don’t understand this.”
Unknotting and the Great Unknown
The conjecture dates back to at least 1937, when the German mathematician Hilmar Wendt set out to understand what happens when you add knots together — that is, when you tie both of them with the same string before gluing the ends together. (Mathematicians call this combined knot the “connect sum.”) Wendt thought that the unknotting number of the resulting knot should always be the sum of the unknotting numbers of the two original knots.
His prediction, now known as the additivity conjecture, makes sense. Say you add the two knots above, whose unknotting numbers are known to be 2 and 3. That means that there’s a sequence of two crossing changes that unknots the lefthand side of the connect sum, and a sequence of three crossing changes that unknots the righthand side. If you use these sequences, you can unknot the whole thing in 2 + 3, or 5, crossing changes.
But this only tells you that the connect sum’s unknotting number is no bigger than 5. You might be able to find a sequence of crossing changes that’s more efficient than untying each side individually. That is, there might be a knot that really is less than the sum of its parts.
To settle the additivity conjecture, mathematicians had to either find a connect sum with a shorter unknotting sequence or prove that no such example exists. In either case, they didn’t have a clue where to begin.
Part of the problem was that the way you lay out your knot — what mathematicians call a “diagram” — determines where and how the knot crosses over itself. There are lots of diagrams that can represent the same knot. To find the shortest sequence of crossing changes, you might have to choose just the right diagram. Often, it’s not the one you’d normally associate with the knot.
“There are unimaginably large numbers of ways to try and imagine changing your diagram before you decide to introduce the crossing change,” Brittenham said. “We don’t, at least at the start, have any control over how complicated the picture has to look.”
In 1985, the mathematician Martin Scharlemann finally made some headway when he proved that for any two knots whose unknotting number is 1, the connect sum will always have an unknotting number of 2. “That made [the whole conjecture] seem much more likely,” said Charles Livingston of Indiana University.
Susan Hermiller (left) and Mark Brittenham disproved a decades-old conjecture about knots, complicating mathematicians’ understanding of these seemingly simple objects.
Courtesy of Susan Hermiller; Courtesy of Mark Brittenham
The result offered tantalizing evidence that the universe of knots could be neatly organized. That’s because all knots can be built out of a smaller class of “prime” knots. The additivity conjecture implied that once you knew the unknotting numbers of those prime knots, you would know them for all knots. Any information you might want about a given knot would fall naturally out of that much simpler set.
Mathematicians wanted the conjecture to be true, said Arunima Ray of the University of Melbourne, “because that would be like, there’s order in the world.”
Scharlemann’s result was later extended to other classes of knots. But it wasn’t clear that it would apply to all knots.
Then Brittenham and Hermiller convened a cluster of computers to help.
Sneakernet
The pair began their project a decade ago with a broader aim: to use computers to learn whatever they could about the unknotting number.
They turned to software known as SnapPy, which uses sophisticated geometric techniques to test whether two pictures depict the same knot. Just a few years earlier, SnapPy had vastly expanded its database, enabling it to identify nearly 60,000 unique knots.
It was perfectly suited for what Brittenham and Hermiller had in mind. They started with a single complicated knot and applied every imaginable crossing change to it, producing scores of new knots. They then used SnapPy to identify those knots — and repeated the process.
They did this for millions of knot diagrams that corresponded to hundreds of thousands of knots. Ultimately, they assembled an enormous library of information about unknotting sequences and calculated upper bounds on the unknotting numbers of thousands of knots. The work required a lot of computing power: The pair signed up for supercomputing time at the University of Nebraska’s computing center, while also running their program on old laptops they’d bought at an auction. All told, they were managing dozens of computers. “We had a bit of a sneakernet,” Brittenham said, “where you transfer information from computer to computer by walking between them.”
Tait tabulated scores of knots and wrote about their properties. This page is from an 1885 paper.
Peter Guthrie Tait
The duo kept their program running in the background for over a decade. During that time, a couple of computers from their ragtag collection succumbed to overheating and even flames. “There was one that actually sent out sparks,” Brittenham said. “That was kind of fun.” (Those machines, he added, were “honorably retired.”)
Then, in the fall of 2024, a paper about a failed attempt to use machine learning to disprove the additivity conjecture caught Brittenham and Hermiller’s attention. Perhaps, they thought, machine learning wasn’t the best approach for this particular problem: If a counterexample to the additivity conjecture was out there, it would be “a needle in a haystack,” Hermiller said. “That’s not quite what things like machine learning are about. They’re about trying to find patterns in things.”
But it reinforced a suspicion the pair already had — that maybe their more carefully honed sneakernet could find the needle.
The Tie That Binds
Brittenham and Hermiller realized they could make use of the unknotting sequences they’d uncovered to look for potential counterexamples to the additivity conjecture.
Imagine again that you have two knots whose unknotting numbers are 2 and 3, and you’re trying to unknot their connect sum. After one crossing change, you get a new knot. If the additivity conjecture is to be believed, then the original knot’s unknotting number should be 5, and this new knot’s should be 4.
But what if this new knot’s unknotting number is already known to be 3? That implies that the original knot can be untied in just four steps, breaking the conjecture.
“We get these middle knots,” Brittenham said. “What can we learn from them?”
He and Hermiller already had the perfect tool for the occasion humming away on their suite of laptops: the database they’d spent the previous decade developing, with its upper bounds on the unknotting numbers of thousands of knots.
The mathematicians started to add pairs of knots and work through the unknotting sequences of their connect sums. They focused on connect sums whose unknotting numbers had only been approximated in the loosest sense, with a big gap between their highest and lowest possible values. But that still left them with a massive list of knots to work through — “definitely in the tens of millions, and probably in the hundreds of millions,” Brittenham said.
For months, their computer program applied crossing changes to these knots and compared the resulting knots to those in their database. One day in late spring, Brittenham checked the program’s output files, as he did most days, to see if anything interesting had turned up. To his great surprise, there was a line of text: “CONNECT SUM BROKEN.” It was a message he and Hermiller had coded into the program — but they’d never expected to actually see it.
Initially, they were doubtful of the result. “The very first thing that went through our heads was there was something wrong with our programming,” Brittenham said.
“We just dropped absolutely everything else,” Hermiller recalled. “All of life just went away. Eating, sleeping got annoying.”
But their program checked out. They even tied the knot it had identified in a rope, then worked through the unknotting procedure by hand, just to make sure.
Their counterexample was real.
Twisted Mysteries
The counterexample Brittenham and Hermiller found is built out of two copies of a knot called the (2, 7) torus knot. This knot is made by winding two strings around each other three and a half times and then gluing their opposing ends together. Its mirror image is made by winding three and a half times in the other direction.
The unknotting number of both the (2, 7) torus knot and its mirror image is 3. But Brittenham and Hermiller’s program found that if you add these knots, you can unknot the result in just five steps — not six, as the additivity conjecture predicted.
“It’s a shockingly simple counterexample,” Moore said. “It goes back to that unpredictability of the crossing change.”
The result led Brittenham and Hermiller to an infinite list of other counterexamples, including almost any knot that’s built by winding two strings and gluing.
Now, with the additivity conjecture decisively struck down, the knot theory community has a wide world to explore.
For some mathematicians, the new result brings disappointment. It reveals that there’s less structure in the world of knots than they had hoped for. The unknotting number is “not as well behaved as we would like,” Ray said. “That’s a bit sad.”
But from another perspective, that only makes the unknotting number more intriguing. “There’s just much more complexity and unknowns about knot theory than we knew there were a few months ago,” Livingston said.
The nature of that additional complexity isn’t clear yet. During their furious examination of their counterexample, Brittenham and Hermiller weren’t able to develop an intuition for why it broke the additivity conjecture when other knots didn’t. Understanding this could help mathematicians get a better handle on what makes some knots complex and others less so.
“I’m still stymied by this most basic question” about the unknotting number, Moore said. “That just lights the fire under you.”
Editor’s Note: Brittenham and Hermiller’s research was funded in part by the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage.
