A student unravels one of the most complex knots in mathematics
The Conway knot, a major knot theory problem, had been unsolved for 50 years. But Lisa Piccirillo found it in less than a week.
When J. H. Conway, the rock star of mathematics, passed away from Covid-19 last April he had at least the consolation that one of the greatest enigmas of the knot theory he had modeled in 1970 was no longer going to be so.
Generations of brilliant mathematicians had racked their brains for decades trying to solve whether their 11-cross knot was slick or not. Slice is one of the properties of knots within this complex branch of mathematics that studies the transformations that can be made in a rope by stretching, twisting, or bending it, but without actually cutting it.
Two years ago, Lisa Piccirillo, a young doctoral student at the University of Austin (Texas), bored during a conference, decided to kill her free time trying to solve this ancient problem. After a few days, she crossed paths with a professor and told him what she had discovered as if nothing had happened.
"Why aren't you excited?" Dr. Cameron Gordon asked, stunned.
This March, the solution to the 11-knot dilemma was published in the prestigious Annals of Mathematics magazine and has taken Piccirillo from a recent graduate to a permanent member of MIT.
"I wasn't allowed to work on the problem during the day, because I didn't consider it to be real math. I thought of it as idle work," the mathematician told Quanta magazine.
"It's something that, let's say, I'm familiar with," she said. "So I just went home and did it.”
The Conway knot is not sliced. But what does that mean?
"In mathematics, a knot would be a rope tied at the ends of which are glued together. The fundamental question we are trying to answer is whether, given two knots, it is possible to get one of them from the deformation of the other. If it is possible, the knots are equivalent," Marithania Silvero, a doctor in mathematics, told El País.
"To solve these questions, knot invariants are used, which are functions that assign a value to each knot. If a given invariant assigns different values to two knots, then it is not possible to deform one knot in the other, that is, they are not equivalent knots," she adds.
Thus, an invariant "studies the properties of a knot, and this knot is only slice if, imagining it in four-dimensional space, it becomes the edge of a disk in this space. Complex, isn't it?
Piccirillo solved the riddle by substituting this Conway knot with another of his invention where the slice property was easier to study. A mixture of creativity and the application of techniques that already existed in the theory of knots.
Are there any loose ends?