Two mathematicians have proved that a straightforward question — how hard is it to untie a knot? — has a complicated answer.

In 1876, Peter Guthrie Tait set out to measure what he called the “beknottedness(opens a new tab)” 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 table and find a spot where the string crosses over itself. Cut the string, swap the positions of the strands, and glue everything back together. This is called a crossing change. If you do this enough times, you’ll be left with an unknotted circle. Tait’s beknottedness is the minimum number of crossing changes that this process requires. Today, it’s known as a knot’s “unknotting number.”

If two knots have different unknotting numbers, then they must be different. But Tait found that his unknotting numbers generated more questions than they answered.

“I have got so thoroughly on one groove,” he wrote in a letter to a friend(opens a new tab), the scientist James Clerk Maxwell, “that I fear I may be missing or unduly exalting something which will appear excessively simple to anyone but myself.”

...read more at quantamagazine.org