Unpicking How to Measure the Complexity of Knots

The original version of this story appeared in Quanta Magazine .

In 1876, Peter Guthrie Tait set out to measure what he called the “ beknottedness ” of knots.

The Scottish mathematician, whose research laid the foundation for modern knot theory, was trying to find a way to tell knots apart—a notoriously difficult task. In math, a knot is a tangled piece of string with its ends glued together. Two knots are the same if you can twist and stretch one into the other without cutting the string. But it’s hard to tell if this is possible based solely on what the knots look like. A knot that seems really complicated and tangled, for instance, might actually be equivalent to a simple loop.

Tait had an idea for how to determine if two knots are different. First, lay a knot flat on a