Graduate Student Solves Decades-Old Conway Knot Problem

It took Lisa Piccirillo less than a week to answer a long-standing question about a strange knot discovered over half a century ago by the legendary John Conway.

By Erica Klarreich

May 19, 2020

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, 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.

“I said, ‘What?? That’s going to the Annals right now!’” Gordon said, referring to Annals of Mathematics, one of the discipline’s top journals.

“He started yelling, ‘Why aren’t you more excited?’” said Piccirillo, now a postdoctoral fellow at Brandeis University. “He sort of freaked out.”

“I don’t think she’d recognized what an old and famous problem this was,” Gordon said.

Piccirillo’s proof appeared in Annals of Mathematics in February. The paper, combined with her other work, has secured her a tenure-track job offer from the Massachusetts Institute of Technology that will begin on July 1, only 14 months after she finished her doctorate.

...

Full article at:

https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/