Unpicking How to Measure the Complexity of Knots

"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.""

Two mathematicians proved that determining how hard it is to untie a knot can be computationally complex. In mathematics, a knot is a looped string with its ends joined; two knots are equivalent if one can be deformed into the other without cutting. Tait defined beknottedness as the minimum number of crossing changes—cutting a crossing, swapping strands, and rejoining—needed to produce an unknotted circle, now called the unknotting number. Different unknotting numbers imply different knots. Despite its descriptive power, the unknotting number resisted classification for 150 years and generated deep, unresolved questions in knot theory.

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