In 1876, Peter Guthrie Tait proposed measuring what he called the 'knottiness' of knots.

The Scottish mathematician, who largely anticipated modern knot theory, was looking for a practical way to distinguish one knot from another—a task that was, to put it mildly, not simple. In mathematics, a knot is a closed loop of string without free ends. Two knots are considered the same if they can be smoothly deformed—pulled, twisted—without cutting, so that one transforms into the other. It is difficult to tell this from a drawing alone: a knot that looks 'terribly' complex might turn out to be the same simple circle.

Tait proposed the following criterion for distinction. Let's lay a knot on a plane and look at its self-intersection points. At one of these points, we 'flip' the crossing: mentally cut it, swap the upper and lower strands, and then 'glue' them back together. By repeating this operation as many times as needed, one can obtain an unknotted circle. He called the minimum number of such 'flips' a measure of unknotting—today, this is known as the unknotting number of a knot.

If two knots have different numbers, they are definitely not equivalent. However, Tait soon realized that the quantity he had introduced created more new mysteries than it solved old ones.

'I have become so engrossed in this subject,' he wrote in a letter to his friend, the physicist James Clerk Maxwell, 'that I fear I might either miss something or, conversely, overestimate what might seem too simple to anyone else.'

If Tait overlooked something, he wasn't alone: for a century and a half, the unknotting number has stumped knot theorists. It is known that in principle it should fully characterize a knot—'perhaps the most basic measure,' as Susan Hermiller of Nebraska says. But in practice, this number is often incredibly difficult to compute, and its connection to the 'complexity' of a knot is not obvious.

To achieve a breakthrough in understanding, in the early 20th century, mathematicians proposed a simple hypothesis about how the unknotting number behaves when two knots are connected (their sum). Proving it would provide a universal way to find this number for any knot, a simple, concrete 'measure' of complexity.

Scottish physicist and mathematician Peter Guthrie Tait began the systematic study of what would become one of the most important problems in knot theory: the classification of knots.

Scottish physicist and mathematician Peter Guthrie Tait began the systematic study of what would become one of the most important problems in knot theory: the classification of knots.

For nearly a hundred years, a proof was sought—unsuccessfully: there were neither confirmations nor refutations.

And then, in a June paper, Hermiller and her co-author Mark Brittenham found a pair of knots whose sum unknots more easily than the hypothesis predicted. This means the hypothesis is false. Moreover, their example allowed for the construction of infinitely many other counterexamples.

'When the paper came out, I literally gasped,' said Allison Moore of Virginia Commonwealth University. According to her, the result shows that the unknotting number behaves 'capriciously and unpredictably,' and the topic is clearly far from being fully understood.

Unknotting and the Great Unknown

Since at least 1937, German mathematician Hilmar Wendt had been trying to understand what happens when two knots are 'added'—by taking a single rope, tying both knots in it, and then joining the ends. Such an object is called a knot sum. Wendt conjectured that the unknotting number of the sum should be equal to the sum of the unknotting numbers of the original knots.

Intuitively, this sounds reasonable. Suppose the left knot has an unknotting number of 2, and the right one has 3. This means there is a sequence of two crossing changes that unknots the left part, and a sequence of three for the right. If performed consecutively, the entire combined knot would be untangled in 2+3=5 steps.

However, this only implies an upper bound: the unknotting number of the sum is no more than 5. It might be possible to find a more clever sequence of changes that untangles the entire knot faster than doing it 'in parts.' In other words, a knot sum could have an unknotting number less than the sum of the numbers of its components.

To confirm additivity, one had to either find an example of a knot sum that untangles faster than 'in parts,' or prove that no such example exists. And this is where everyone got stuck—it was unclear where to even begin.

One reason is that a knot on a plane is defined by a diagram, and the same knot can have countless diagrams. The choice of diagram determines where the crossings are located. To find the shortest sequence of crossing 'flips,' it is often necessary to first select a suitable diagram—and this is not always the familiar image of the knot.

'The number of ways to modify a diagram before applying a crossing change is unimaginably large,' says Mark Brittenham. 'At least at the beginning, we have no control over how complex the required diagram might be.'

In 1985, Martin Scharlemann took the first significant step: he proved that if both knots have an unknotting number of 1, then their sum always has an unknotting number of 2. 'This greatly increased the plausibility of the whole hypothesis,' notes Charles Livingston of Indiana University.

Susan Hermiller (left) and Mark Brittenham disproved a decades-old knot hypothesis, complicating mathematicians' understanding of these seemingly simple objects.

Susan Hermiller (left) and Mark Brittenham disproved a decades-old knot hypothesis, complicating mathematicians' understanding of these seemingly simple objects.

The results of that work inspired hope that the 'world of knots' could be ordered. The fact is that any knot can be constructed from a smaller set of basic, 'prime' knots. And if the additivity hypothesis were true, then knowing the unknotting numbers for these simple building blocks would automatically give us the numbers for all composite knots: the necessary information would 'flow' from a small dataset to all objects.

'We really wanted the hypothesis to be correct,' notes Arunima Ray of Melbourne, 'because it would have meant that there is order in this field.'

Later, Scharlemann's result was extended to a few more families of knots, but it remained unclear whether it covered all cases. Then, Brittenham and Hermiller brought in computational power—using several computers and moving to large-scale brute-force searches.

The Hidden Internet

About ten years ago, Hermiller and Brittenham launched a major project: to gather as much information as possible about the unknotting number through computation.

To do this, they used the program SnapPy, which relies on sophisticated geometric methods to check if two diagrams correspond to the same knot. Over several years, the SnapPy database grew significantly and learned to recognize almost 60,000 different knots—exactly what they needed.

Their approach was as follows: take one complex knot and perform all possible crossing changes one by one, generating a whole family of new knots. Then, use SnapPy to identify each of them—and repeat the process.

In this way, they processed millions of diagrams covering hundreds of thousands of knots, compiling a huge database of 'unknotting' sequences and calculating upper bounds for the unknotting number in thousands of cases. The project required significant resources: the scientists connected to the supercomputers at the University of Nebraska and also ran code on used laptops purchased at auctions. In the end, they were managing dozens of machines. 'We ended up with almost a home computing network,' recalls Brittenham, 'where data was literally carried from one computer to another by walking between them.'

Tait compiled a table of knots and described their properties. This page is from an 1885 paper.

Tait compiled a table of knots and described their properties. This page is from an 1885 paper.

For over ten years, their program ran in the background. During this time, some of the mismatched computers overheated and even caught fire. 'One was sparking,' Brittenham recalls. 'It was amusing.' (Eventually, all the 'veterans' were honorably retired.)

In the fall of 2024, Brittenham and Hermiller's attention was drawn to a paper about an unsuccessful attempt to disprove additivity using machine learning. They decided that ML was not the best tool for such a task: if a counterexample exists, it would be a real 'needle in a haystack,' and machine learning is better at finding patterns than rare exceptions. However, this work reinforced their belief that their own well-designed computing network was just the thing to find such a needle.

Knots That 'Bind'

The researchers figured out how to use their collected unknotting sequences to hunt for possible counterexamples to additivity.

Imagine two knots with unknotting numbers of 2 and 3; we are considering their sum. By making one crossing change, we get an 'intermediate' knot. If additivity holds, the original sum should require 5 steps, and the new one, after the first step, should have 4 remaining.

But if it is already known that this intermediate knot can be unknotted in 3 steps, then the original sum can be untangled in 4 steps instead of 5—and the hypothesis collapses.

'We have these intermediate knots,' says Brittenham. 'What can they teach us?'

And this is where their decade-old database came in handy: a huge catalog of upper bounds for the unknotting numbers of thousands of knots, already running on their laptops.

The scientists began 'adding' knots in pairs and searching for the shortest sequences to unknot them. They prioritized sums for which only rough estimates were known—those with a large gap between the lower and upper bounds of the unknotting number. Even so, the list of candidates was enormous—'tens of millions, maybe even hundreds of millions,' according to Brittenham.

For several months, their program systematically changed crossings, checking the resulting knots against their own database. Then, late one spring, Brittenham was checking the output files as usual—and saw the line: 'Connect sum violated.' They had included this message in the code in case a counterexample was found, but never expected to see it.

At first, they didn't believe it. 'The first thought was that we had a bug in the program,' recalls Brittenham. 'We dropped everything,' says Hermiller. 'Eating and sleeping just got in the way.'

However, a check confirmed it: everything was correct. They even tied the corresponding knot in a rope and untangled it by hand—just to be completely sure.

The counterexample was real.

The counterexample found by Brittenham and Hermiller is the sum of two copies of the torus knot T(2,7). Such a knot can be obtained by taking two strands and giving them seven half-twists (i.e., wrapping them around each other three and a half times) and then joining the ends; the mirror image is obtained by 'twisting' in the opposite direction.

Both T(2,7) and its mirror image have an unknotting number of 3. But their sum, as the researchers' program showed, can be unknotted not in 6 steps, as additivity would predict, but in just 5—that is, faster than expected.

'The counterexample is surprisingly simple,' notes Moore. 'It all comes down to the unpredictability of a single step—a crossing change.'

Based on their finding, Brittenham and Hermiller constructed an infinite family of new counterexamples—effectively covering almost all knots obtained by winding two strands and then gluing them.

Now that the additivity hypothesis has been definitively disproven, knot theory faces a whole new set of questions and directions. This is disappointing for some: there is less structure in the 'world of knots' than hoped for. 'The unknotting number doesn't behave as nicely as we had hoped,' says Ray. 'It's a little sad.'

But at the same time, the intrigue only grows. 'It seems knot theory is much more intricate and rich with unknowns than we thought even very recently,' says Livingston.

It's still unclear what this additional complexity consists of. After carefully studying their example, Brittenham and Hermiller couldn't explain why it breaks additivity while others do not. By figuring this out, mathematicians might understand what makes some knots 'more complex' and others simpler.

'I still can't answer this seemingly simple question about the unknotting number,' admits Moore. 'And that only fuels the interest.'