For over a century, a seemingly simple conjecture in knot theory—the additivity conjecture—remained stubbornly unproven. This conjecture, proposed to simplify the measurement of knot complexity using the 'unknotting number,' posited that the unknotting number of two knots combined is simply the sum of their individual unknotting numbers. This elegant idea, dating back to at least 1937, offered the promise of a straightforward method to classify the intricate world of knots.
The unknotting number represents the minimum number of crossing changes needed to transform a knotted string into an unknotted loop. While intuitively appealing, calculating this number for even moderately complex knots presents a formidable challenge. The additivity conjecture aimed to circumvent this difficulty by providing a shortcut: simply add the unknotting numbers of the constituent knots to determine the combined knot's complexity.
However, a recent groundbreaking paper by Susan Hermiller and Mark Brittenham has overturned this long-held belief. Using a combination of sophisticated software (SnapPy), extensive computational resources, and a decade-long, painstaking process involving hundreds of thousands of knot analyses, they discovered a counterexample—a pair of knots whose combined unknotting number defied the conjecture's prediction.
Their approach involved generating a vast database of knot information using SnapPy and then systematically searching for instances where the combined unknotting number was less than the sum of the individual unknotting numbers. This involved processing millions of knot diagrams and running their program across dozens of computers, including older repurposed laptops. The discovery was greeted with astonishment by the mathematical community, with one expert exclaiming, "When the paper was posted, I gasped out loud."
The counterexample consists of two copies of a (2, 7) torus knot and its mirror image, each having an unknotting number of 3. Intriguingly, their connect sum—the knot formed by joining them—can be unknotted in just five steps, contradicting the conjecture's prediction of six steps. This seemingly simple counterexample opened the door to an infinite family of similar counterexamples.
The implications of this discovery are profound. While initially met with some disappointment among mathematicians who valued the order the conjecture seemed to imply, the refutation of the additivity conjecture instead reveals a deeper, more chaotic and complex reality within knot theory. The unexpected result shines a spotlight on the inherent unpredictability of the unknotting number, opening up new avenues for research and highlighting the vast unknown terrain that remains to be explored within this fascinating field. The work emphasizes the critical role of computational power and persistence in tackling significant mathematical problems.
---
Originally published at: https://www.quantamagazine.org/a-simple-way-to-measure-knots-has-come-unraveled-20250922/
