A Simple Way To Measure Knots Has Come Unraveled Author: Leila Sloman Date: September 22, 2025 Topics: Knot Theory, Mathematics, Topology --- Overview Mathematicians Susan Hermiller and Mark Brittenham have disproven a longstanding conjecture in knot theory regarding how the complexity of knots combines under a connect sum operation. The conjecture, known as the additivity conjecture, suggested the unknotting number (a measure of how difficult it is to untie a knot) of the combined knot equals the sum of the unknotting numbers of the component knots. Their discovery reveals the unknotting number is more chaotic and less predictable than previously thought, complicating understanding of knot complexity. --- Background: What is Knot Theory and Unknotting Number? A knot is modeled as a tangled loop of string with glued ends. Two knots are considered the same if one can be deformed into the other without cutting. The unknotting number is the minimum number of crossing changes (switching which strand goes over/under at a crossing) required to transform a knot into a simple untangled loop. Peter Guthrie Tait introduced the concept in the 19th century, originally called "beknottedness," but many mathematicians have struggled with its complexity and computability. --- The Additivity Conjecture Proposed around 1937 by Hilmar Wendt. States that the unknotting number for a connect sum (tying two knots on the same string before closing it) is the sum of the unknotting numbers of the individual knots. Although intuitive (untie each knot separately), this only guarantees an upper bound, leaving open whether fewer crossing changes might exist to untie the combined knot. Martin Scharlemann proved the conjecture true for knots with unknotting number 1. This gave hope of an orderly classification. --- Hermiller & Brittenham's Approach Spent over a decade collecting data on unknotting sequences using SnapPy software, which identifies and studies knots using geometry. They ran millions of computer simulations applying crossing changes to knots, creating a vast database of unknotting information. The team set up a distributed computing system—"sneakernet"—moving data physically among computers to manage the workload. Motivated by a failed machine learning approach, they focused on exhaustive computational searches instead of pattern recognition. --- Disproving the Conjecture Using their dataset, they searched for counterexamples by: Combining pairs of knots, Applying crossing changes, Comparing resulting knots’ unknotting numbers with known values. Discovered a "middle knot" with a lower unknotting number than predicted by additivity, disproving the conjecture. Verified the finding physically by tying the knot and untying it by hand. Found the counterexample involves combined (2,7) torus knots and their mirror images, each with unknotting number 3, but their sum can be undone in just 5 crossing changes, not 6. --- Significance and Reactions The result reveals the unknotting number is less structured and more unpredictable. It's a surprising and simple-to-express counterexample revealing more complexity in knot theory than expected. Some mathematicians feel disappointed as it implies less order in knot structure. Others are excited as it opens new avenues for understanding knots' complexity and behavior. The discovery suggests infinite families of similar counterexamples exist. --- Related and Further Reading “Untangling Why Knots Are Important” “Graduate Student Solves Decades-Old Conway Knot Problem” “Mathematicians Eliminate Long-Standing Threat to Knot Conjecture” --- Additional Notes The research was partially funded by the Simons Foundation, an editorially independent supporter of Quanta Magazine. Quanta Magazine invites readers to subscribe to newsletters, explore related mathematics content,