Solving Knotty Problems
Matthew Hedden, assistant professor of mathematics at MSU, has a passion for knots.
While most of us see a knotty problem as . . . well, a problem . . . for Hedden, it’s a welcome challenge. Hedden’s research interest is low-dimensional topology. More specifically, a branch of his research is focused on knot theory—the study of knots in a mathematical sense.
To a mathematician, a knot is a closed piece of string with no loose ends. Or, in mathematical parlance, the “embedding of a circle in three-dimensional Euclidean space.” It is quite unlike the definition of a knot in everyday life, such as in a shoelace or a rope, which has two loose ends.
“One of the things I find fascinating about knot theory is that it connects with a lot of other areas of mathematics . . . Knots are the building blocks for all three- and four-dimensional shapes,” Hedden said.
“Knots are also relevant from the perspective of trying to find meaningful models of our universe. In fact, some of the most powerful tools for studying knots come out of theoretical physics.
“For instance, if we lived in a pretzel dough universe, and if I were to remove a pretzel-shaped knot, there would be a pretzel-shaped hole there. If I were to replace it, but twist it, and glue it back into the hole, it would be different. This would change the shape of the universe.”
While Hedden is a “pure” mathematician, there are emerging applications for knot theory—such as in biology. For example, the DNA of various bacteria can become “knotted.” In order to replicate, the DNA must “unknot” itself. Specific enzymes, known as topoisomerases, perform the act of “cutting” the string (or circular stand of DNA) and passing it through itself, to accomplish the unknotting. In this case, the complexity of the knots becomes highly relevant. The number of times the string needs to be cut is known as the unknotting number.
Perhaps the most fundamental question about knots is: Is there an algorithm, or is there some way to determine whether or not what appears to be a “knotted up” piece of string is actually an unknot? In a short paper published in 2009, Hedden demonstrated the use of the Khovanov homology theory, which was developed in the late 1990s, to detect an unknot in an algorithmic way.
Unlike number theory or Euclidean geometry, which are of course very old, the field of general topology is fairly new. While there was some work on knot theory in the late 1800s, it didn’t become popular until the second half of the last century. Indeed, Hedden’s particular area of research, low-dimensional topology, became popular even more recently—in the 1970s and 1980s.
“This is typically how new areas of math emerge. Somebody has a great new idea, a great new perspective on something. And in the field of mathematics, there is always a zoo of new ideas and perspectives,” noted Hedden.
“I don’t know any area of mathematics where having some topological intuition isn’t helpful. It’s a framework that is useful for every branch of math,” Hedden added.
Today, topology (along with analysis and algebra) is one of three core areas in most pure math doctoral programs in math departments at universities across the United States.
So, will Hedden ever tire of his research work? Knot a chance!
— Val Osowski, College of Natural Science
— Photo of Dr. Hedden by Harley Seeley