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Graduate Student Untangles Decades-Old Math Problem in Less Than a Week

Lisa Piccirillo recently published her proof of Conway’s knot problem, a well-known quandry that stumped mathematicians for more than 50 years

The Conway Knot (Illustration by Saung Tadashi, public domain via Wikimedia Commons)
smithsonianmag.com

Conway’s knot, a famous mathematical problem, was a tricky one to untangle. Mathematicians have been arguing about how to solve it for more than 50 years—until 2018, when graduate student Lisa Piccirillo took it up on a whim and figured it out in less than a week, Erica Klarreich reports for Quanta magazine.

Piccirillo, who recently published her proof in the Annals of Mathematics, was a graduate student at the University of Texas Austin when she cracked the problem. She first learned of it at a conference in the summer of 2018, and spent her evenings working on it until she had her answer. Solving the problem took Piccirillo less than a week, per Quanta.

Piccirillo didn’t realize just how exciting her find was until she shared it with a professor at UT Austin. “He started yelling, ‘Why aren’t you more excited?’” Piccirillo tells Quanta. “He sort of freaked out.”

As Caroline Delbert reports for Popular Mechanics, a mathematical knot is similar to a twisting normal knot—in a tangled necklace or a shoelace, for example—except both ends of the knot are connected in a circle. Knot theory, a field of topology, is the mathematical study of these kinds of snarls, per Wolfram Alpha. Knot theory has helped enhance our understanding of the shape of DNA and the possible form of the universe, Erin Blakemore reports for the Washington Post.

The Conway knot is a mathematical knot with 11 crossings discovered by mathematician John Horton Conway. The knot is so famous that it decorates the gates of the Isaac Newton Institute for Mathematical Sciences at Cambridge University, per the Washington Post.

Its “problem” is a question that has persisted for decades: is the Conway knot a slice of a higher-dimensional knot? A knot that is “slice” is one that can be made by slicing a knotted sphere in four-dimensional space, per Quanta.

Now, Piccirillo has an answer: the Conway knot is not “slice.” Piccirillo, who has since landed a tenure-track position at MIT, figured out the problem by studying the knot’s “trace,” a four-dimensional shape associated with each knot. Some knots are “trace siblings,” meaning they have the same four-dimensional pattern. Mathematicians know that trace siblings have the same “slice status,” Klarreich explains.

Piccirillo found the trace of Conway’s knot, then constructed another complicated knot—now called Piccirillo’s knot—that had the same trace as Conway’s knot. Because Picirillo’s knot is not slice, Conway’s knot is not slice either.

Conway, the mathematician who first discovered the 11-crossing knot, died at 82 years old last month due to complications of COVID-19. As Siobhan Roberts writes in Conway’s obituary in the New York Times, Conway was a world-famous mathematician known for his playful teaching style and contributions to the field.

“This question, whether the Conway knot is slice, had been kind of a touchstone for a lot of the modern developments around the general area of knot theory,” Joshua Greene, a mathematician at Boston College who supervised Piccirillo’s undergraduate senior thesis, tells Quanta. “It was really gratifying to see somebody I’d known for so long suddenly pull the sword from the stone.”

About Nora McGreevy

Nora McGreevy is a freelance journalist based in South Bend, Indiana. Her work has appeared in Wired, Washingtonian, the Boston Globe, South Bend Tribune, the New York Times and more. She can be reached through her website, noramcgreevy.com.

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