math

In 1876, Peter Guthrie Tait set out to measure what he called the “beknottedness” of knots.

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.

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 (opens a new tab) 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 (opens a new tab) 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 (opens a new tab) — and used their counterexample to find infinitely many other pairs of knots that also disprove it.

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.”