Now it’s been solved by a graduate student who tackled it in her spare time — and figured it out in less than a week.
The problem is one of the long-standing mysteries of knot theory, which is a branch of topology, a kind of geometry that looks at the nature of spaces. The knots in question look like tangled loops, and they’ve contributed to researchers’ understanding of everything from the shape of DNA to the behavior of economic markets to the possible form of the universe.
Their study isn’t limited to one dimension, either: Mathematicians also consider how knots made of two-dimensional spheres behave in four-dimensional space. The fourth dimension offers unlimited space in which to unravel even the most complicated knot.
“It’s hard to visualize a knotted sphere in 4D space, but it helps to first think about an ordinary sphere in 3D space,” journalist Erica Klarreich explains in a fascinating feature on the now-solved problem for Quanta magazine. “If you slice through it, you’ll see an unknotted loop. But when you slice through a knotted sphere in 4D space, you might see a knotted loop instead (or possibly an unknotted loop or a link of several loops, depending on where you slice). Any knot you can make by slicing a knotted sphere is said to be ‘slice.’ ”
Mathematicians have asked themselves for years whether Conway’s knot, which has 11 crossings, is slice, but it took Lisa Piccirillo, then a graduate student, less than a week to solve the problem using a creative strategy. Her proof has been hailed as a thing of mathematical beauty, and her work could point to new ways to understand knots, too.
Klarreich’s report, and her savvy explanation of math that might seem mind-boggling otherwise, is worth a read. Check out the article at bit.ly/conwayknot.