Mark Brittenham and Susan Hermiller disproved a fundamental, 88-year-old conjecture in knot theory—that linking two knots together can actually produce a knot that is easier to untie.
In knot theory, people study knots — loops in space that don’t have any ends. One important idea in knot theory is the unknotting number, which is how many changes you need to make to turn a knot into a simple loop (called the unknot). The fewer changes you need, the simpler the knot.
Combining two knots is called a connected sum — you cut a small piece out of each knot and glue them together to make a new one. People thought that if you combine two knots, the number of changes you'd need to untie the new knot would just be the sum of the two original numbers. For example, if one knot takes 3 changes and another takes 2, the combined one should take 5—called the additivity conjecture.
But Brittenham and Hermiller found out that this isn’t always true. They took a knot called the (2,7)-torus knot and joined it with its mirror image. It should take 6 changes to untie, but it actually takes only 5 or fewer, meaning unknotting numbers don’t always add up the way as originally thought.
"It was very unexpected and very surprising," Brittenham, professor in the Department of Mathematics, said.
Their work is covered in New Scientist (subscription required), and will be further shared in Quanta and Scientific American. The original study was published on Cornell University's arXiv.
The result, which professor and department chair Petronela Radu calls "an amazing accomplishment," is the culmination of years of work.
