How many nodes were discovered? A clever algorithm solves the problem

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Finding out how many knots there are for a given number of string crossings has been thought to be an incredibly complex task. Now the algorithmics cracks it – and shows us how to solve other fiendishly complicated math problems

mathematics


July 12, 2022


Spencer Wilson

It used to be one of the most frustrating parts of any public transit journey. You squeeze past the other bodies, sit down, and fish your headphones out of your pocket. You didn’t bother to coil the wires into a neat loop last time, so – sigh – you now have to spend the next 5 minutes untangling that knot. Thank god for inventing wireless earbuds.

However, knots are not just an everyday annoyance. They are also a source of endless inspiration for researchers. Consider mathematician Benjamin Burton, who is fascinated by a simple question: How many knots are there? “There’s something exciting about problems that you can describe to a 10-year-old that mathematicians haven’t solved yet,” he says.

Keeping a knot count is one of those problems that should be impossible to solve due to its complexity. There are so many ways the strings can be crossed and looped that even the fastest computer could never catalog them all. But Burton tried, and along the way showed that with a few clever arithmetic tricks, many math problems that seem unsolvable may not be.

Knots and science have been, um, entangled for quite some time. In the last decades of the 19th century, scientists struggled to understand atoms. One hypothesis saw them as small vortices of fluid that became stable through knotting. Lord Kelvin, who later became President of the British Royal Society, was the first to propose that each chemical element…

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