With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots.

From the tangle in your computer cord to the mess your cat made of your knitting basket, knots are everywhere in daily life. They also pervade science, showing up in loops of DNA, intertwined polymer strands, and swirling water currents. And within pure mathematics, knots are the key to many central questions in topology.

Yet knot theorists still struggle with the most basic of questions: how to tell two knots apart.

It’s hard to decide whether two complicated knots have the same structure just by looking at them. Even if they appear completely different, you might be able to turn one into the other by moving some strands around. (To a mathematician, the ends of a knot are always fastened together so that such motions won’t untie it.)

Over the past century, knot theorists have developed a set of clear, if imperfect, tools for distinguishing knots. Called knot invariants, these tools each measure some aspect of a knot — a pattern formed by its interwoven strands, perhaps, or the topology of the space surrounding it. If you use an invariant to measure two knots and you get two different results, you’ve proved the knots are different. But the reverse isn’t always true: If the invariant gives you identical results, the knots may be the same, or they may be different.

Some invariants are better at telling knots apart than others, but there’s a trade-off: These stronger invariants tend to be hard to calculate. “Most invariants are either very strong but impossible to compute, or easy to compute but very weak,” said Daniel Tubbenhauer of the University of Sydney.













...read more at quantamagazine.org