A Grad Student Solved the Epic Conway Knot Problem—in a Week
Lisa Piccirillo encountered the more than 50-year-old question by chance at a conference. Her solution relies on a classical tool called the knot trace.
IN THE SUMMER of 2018, at a conference on low-dimensional topology and geometry, Lisa Piccirillo heard about a nice little math problem. It seemed like a good testing ground for some techniques she had been developing as a graduate student at the University of Texas at Austin.
“I didn’t allow myself to work on it during the day,” she said, “because I didn’t consider it to be real math. I thought it was, like, my homework.”
The question asked whether the Conway knot—a snarl discovered more than half a century ago by the legendary mathematician John Horton Conway—is a slice of a higher-dimensional knot. “Sliceness” is one of the first natural questions knot theorists ask about knots in higher-dimensional spaces, and mathematicians had been able to answer it for all of the thousands of knots with 12 or fewer crossings—except one. The Conway knot, which has 11 crossings, had thumbed its nose at mathematicians for decades.
Before the week was out, Piccirillo had an answer: The Conway knot is not “slice.” A few days later, she met with Cameron Gordon, a professor at UT Austin, and casually mentioned her solution.