GoKawiilFrom 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.
By the time you’re up to knots whose strands cross each other 15 or 20 times, many invariants start to falter — either they fail to distinguish between many knots, or they’re getting too hard to compute. For most knot invariants, said Dror Bar-Natan of the University of Toronto, “if you say ‘300 crossings’ and then you say the word ‘compute,’ you are in science fiction.”
A page from an 1885 paper by Peter Guthrie Tait, in which he distinguishes different knots with 10 crossings. Peter Guthrie Tait
But now, Bar-Natan and Roland van der Veen of the University of Groningen in the Netherlands have come up with a knot invariant that does not require mathematicians to choose between two evils: It is both strong and easy to compute. “It seems to be right in the sweet spot where exciting things happen,” said Tubbenhauer, who was not involved in the work.
This combination of strength and speed means that mathematicians can probe knots that were previously far out of reach. It’s easy to calculate the new invariant for knots with as many as 300 crossings, and Bar-Natan and van der Veen have even calculated some aspects of the invariant for knots with more than 600 crossings.
We in some sense just winged it. Roland van der Veen, University of Groningen
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