Knotty Math Comes Undone

Knots hold a special place in the hearts of sailors and mathematicians.

That’s almost self-explanatory for sailors, who have been tying knots as an essential part of their craft for centuries. They even used them to tell how fast they’re going over open water.

Mathematicians’ fascination with knots is harder to explain, but comes down to their love of counting. The more complicated the knot, the more the knots strands cross each other, the better. And what better way to make a knot more complicated than by taking two knots, cutting each and then joining their open ends together to make an even bigger, more complicated knot?

They even defined a special rule about the practice to calculate how much more complicated the resulting knot would be. They conjectured that if they took the unknotting number for each knot, which is to say the number of steps it would take to tranform the knot into a simple loop (imaginatively called the “unknot”), and added them together, the result would be the unknotting number of the combined, more complicated knot.

This additivity conjecture worked with just about every combination of knots they could throw at it. Until two mathematicians discovered an example where that rule didn’t work. Instead of leading to a more complicated knot, they found a knot that became easier to unknot after being joined. It didn’t add up as previous generations of mathematicians had believed it should, and because it didn’t, they proved the conjecture about the additivity of unknotting numbers is untrue for all cases.

In the following video, Trefor Bazett guides viewers through basic knot theory in an easy to understand presentation before getting to the remarkable disproof of the additivity conjecture in knot theory by Mark Brittenham and Susan Hermiller, which only took them ten years to work out:

Brittenham and Hermiller’s paper is here. Bazett has a second video featuring an extended interview with Brittenham and Hermiller.

The knot they found that broke the rule is the (2,7) Torus Knot, an example of which is the featured illustration for this article. It’s also known as the 7₁ knot in a different mathematical knot nomenclature system, but we’re not going to get into that topic because why complicate knots any more than needed?

Image credit: Blue 7₁ Knot or (2,7) Torus Knot by Jim.belk on Wikimedia Commons. Public Domain.