How Complex Is a Knot? New Proof Reveals Ranking System That Works.

Back in 1981, Cameron Gordon introduced a new way to relate two knots — mathematical constructs modeled after the knots that appear in a single thread or string. In his paper, he conjectured that this new relationship could be used to arrange groups of knots according to how complicated they are.

This winter, Ian Agol, a mathematician at the University of California, Berkeley, posted a six-page paper that proved Gordon’s conjecture, giving mathematicians a new way to order knots by complexity. “What was really surprising about this paper is, one, that it’s super short,” said Arunima Ray, a researcher at the Max Planck Institute for Mathematics. “And secondly, that it’s using some tools that are, let’s say, unusual to this particular question.”

Progress on the question was slow until 2019, when Ian Zemke, a mathematician at Princeton University, showed how to apply powerful new methods to the problem that weren’t around in the 1980s. Agol learned about the conjecture a little over a year ago, and he started working on it in earnest late last fall.

“That was nice, other people were kind of thinking about this problem,” said Gordon, who is a professor at the University of Texas, Austin. “And then, somewhat out of the blue, along comes Ian Agol with his beautifully short, beautifully elegant proof.”

Gordon’s conjecture is one of many in knot theory that attempt to organize the infinitely tangled universe of knots. At the heart of this project is the observation that you can drastically alter a knot’s appearance by twisting the strands or sliding them around. (To prevent mathematicians from simply unraveling the string and retying it however they like, the ends of the string are merged to form a closed loop, like a rubber band.) Given drawings of knots, knot theorists try to figure out which ones are truly distinct, and which are different depictions of the same object.

“You throw [two knots] down on the table and they may look completely different, right?” said Gordon. “But if you can move one around so that it looks exactly like the other, then you say the knots are the same.”

Figuring out which knots are equivalent is extremely difficult. A simple loop can easily be twisted up and unrecognizable. And two complex knots might superficially appear to be equivalent, even when they’re not.

In many cases, mathematicians resort to a less stringent concept called concordance. Concordance involves situating your three-dimensional knot in four-dimensional space. When mathematicians add an extra dimension, new ways of contorting knots suddenly open up.

Think of how you’re confined to the two-dimensional ground. You can only walk forward and backward, or from side to side (or some combination of these). If you happen to be surrounded by a giant brick wall, you’ll be stuck. But an eagle doesn’t have this problem. It can fly up into the third dimension, so a wall can’t contain it.

In the same way, a fourth dimension gives a knot a lot more leeway — too much, in fact. Any knot, no matter how tangled, can be untied in a four-dimensional space. So mathematicians add a rule: Two knots are concordant if they can be connected by a certain kind of imaginary cylinder.

Suppose you have two unknotted rubber bands. Now suppose you roll up a sheet of paper into a cylinder and slip a rubber band around each end. The paper cylinder connects the rubber bands.

But if you switched out one of the rubber bands for a defective one which had become irrevocably knotted up in manufacturing, you’d struggle. A cylinder with an edge that matched this tangled rubber band would inevitably cross through itself at some point. In math-speak, it wouldn’t be “smooth.” If you moved your cylinder into four dimensions, however, you might be able to smooth it out. Two knots are concordant when they can be connected by a smooth cylinder in four-dimensional space.

Four decades ago, Gordon suggested a way to compare two knots by complexity, based on concordance. To do so, he had to apply one more restriction to the cylinders. In ordinary concordance, mathematicians allow themselves some seemingly questionable moves to make things easier, such as pinching off loops or attaching extra ones. Gordon’s adaptation, which he called ribbon concordance, doesn’t allow you to add extra loops.

This restriction has an important consequence: Unlike a regular concordance, ribbon concordance doesn’t necessarily go both ways. If you have two knots, you might have a ribbon concordance that can transform the first knot into the second, but not the other way around.

Gordon thought that this property could be used to rank the knots against each other. If you need to add loops to the second knot in order to transform it into the first, it is effectively a less complicated knot. That creates an ordering of any set of knots that have ribbon concordances between them.

To show that ribbon concordance has the crucial properties that we expect of orderings, mathematicians had to prove three things. The first two are obvious, and Gordon pointed them out in his original paper. To finish the proof, Agol needed to show that if two knots are both ribbon concordant to each other, they have to be equal.

Late one night, shortly after Agol began to study the problem, he had a flash of insight. There was a connection between the conjecture and an answer he’d recently posted on the math discussion website MathOverflow. “[Someone] asked this question and I happened to know the tools to answer it,” Agol said. “Then I was realizing that this might be relevant to this problem.”

The following day, Agol felt more pessimistic. “A lot of times you think you’ve solved something, you get excited for a day or something until you find some mistake,” he said. That may seem like a particular danger here, as Agol is something of an outsider to the area. He typically studies objects in three-dimensional space, not 4D. But colleagues think Agol’s fresh eyes may have worked in his favor. “A lot of mathematicians are very siloed,” said Tye Lidman, a professor at North Carolina State University. “I think something that Ian does really well is he is going around and seeing what’s going on in pretty much every area and knows something about every area.”

In the end, Agol didn’t use any of the modern techniques that had triggered the problem’s surge in popularity. Instead, he studied collections of equations that encode information about the knots, using ideas that have been around for decades.

Gordon was surprised by the simplicity of the solution.

“The techniques he used, it could have been proved at the time I wrote the paper, really. So it makes it even more interesting,” he said.

Editor’s note: Ian Agol has received research funding from the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage.