Over 60 years ago, Ralph Fox posed a problem about knots that haunts mathematicians to this day. His question is now often formulated as the “slice-ribbon conjecture,” which posits that two seemingly distinct groups of knots are actually the same. With its suggestion of elegant simplicity within the world of knots, it’s become one of the most high-profile problems in knot theory. “It would mean that the world is a little bit more structured than you might expect otherwise,” said Arunima Ray, a mathematician at the Max Planck Institute for Mathematics in Bonn.
For decades, one particular knot was suspected to be a possible route to settling the conjecture. Yet in a paper posted last summer, five mathematicians found that this knot isn’t going to work after all. While the arguments they introduced will provide new insights into a broader class of knots, the work as a whole leaves mathematicians uncertain about the conjecture. “I think there’s actual legitimate controversy on whether it’s going to turn out to be true or not,” said Kristen Hendricks, a mathematician at Rutgers University.
The slice-ribbon conjecture concerns two types of knots: slice knots and ribbon knots. Figuring out which knots are slice is “one of the fundamental questions that our subject revolves around,” said Abhishek Mallick, one of the authors of the new paper.
A mathematical knot can be thought of as an ordinary loop of string. Mathematicians call a simple loop without a knot in it the “unknot.” (Though this isn’t a knot in the ordinary sense of the word, mathematicians think of the unknot as the simplest example of a knot.)
Knots also define the boundary of a shape that mathematicians call a disk, even though it doesn’t always look disk-like in the ordinary sense of the word. The simplest example, the unknot, forms the boundary of a circle — a “disk” that does indeed look like a disk. But the loop forms the boundary not only of a circle that lies flat on a table, but also of a bowl — which extends into three dimensions — that’s laid upside down on top of the table. The disks that knots define can be further extended from three dimensions into four.
If there is a knot in the string, the disks get more complicated. In three-dimensional space, those disks have singularities — points where they are mathematically ill-behaved. Slice knots are those for which it’s possible — in four dimensions — to find a disk without such singularities. Slice knots are the “next-best thing to the unknot,” as Peter Teichner, also of the Max Planck Institute, has put it.
Despite that, the disks bounded by slice knots in three dim
