New Knot Theory Discovery Overturns Long-Held Mathematical Assumption

Summary

September 2, 20252 min readNew Knot Theory Discovery Overturns Long-Held Mathematical AssumptionMathematicians have unraveled a key conjecture about knot theoryBy Max Springer edited by Sarah Lewin FrasierIn a recent preprint paper, mathematicians connected two knots in a way that could be undone in a surprisingly small number of moves. Scanning the crowd at a fancy soiree may reveal a wide array of neckties, each fastened with a highly complex mathematical object masquerading as fashion. An entire field of mathematics is devoted to understanding mathematical knots, which one can obtain from any traditional knot by gluing the loose ends together. Mathematicians long believed that if you attach cut ends of two different knots to each other, the new knot will be just as complex as the sum of the individual knots’ complexity. But researchers recently managed to find a knot that is simpler than the sum of its parts.Knot theory is a branch of topology that has surprisingly practical applications, such as understanding how proteins coil DNA and how molecular structures remain stable. The theory’s central question: How can we tell which knots are unique or which are the same as others? Mathematicians consider two knots the same if one can be manipulated to look like the other without being cut open—any knots you can produce with mere tugging and pulling are fundamentally the same. Only cutting and reconnecting to let two strands cross yields unique knots.Using these careful manipulations, mathematicians assign each knot an unknotting number, which is the minimum number of cutting and reconnecting “moves” it would take to unravel the knot into a simple loop. This computation is often deceptively difficult. Many mathematicians assumed that if we construct a larger knot by joining together smaller ones whose unknotting numbers are known, then the quickest way to untangle the larger knot will be by simply undoing each piece independently. This idea that two conjoined knots’ un...