# Knot Theory

Posted on: October 23, 2010

A few days ago, Dr. Daniel Heath, a math professor at Pacific Lutheran University, presented a lecture during Capstone Seminar offering an introduction to knot theory. He will be teaching a special topics class in the spring, and was hoping to pique our interest in the subject. Knot theory is a branch of topology in which a circle is embedded in 3-dimensional Euclidean space, $\mathbb{R}^3$. According to the Oracle ThinkQuest site, “a knot is a closed, one dimensional, and non-intersecting curve in three-dimensional space. From a more mathematical and set-theoretic standpoint, a knot is a homeomorphism that maps a circle into three-dimensional space and cannot be reduced to the unknot by an ambient isotopy.” So what you have is something like a knot in a string, but with the ends joined together so that it cannot come undone. There are various ways to describe a knot, so an important problem in knot theory is determining when two different representations are describing the same knot.  Wikipedia explains: “Two mathematical knots are equivalent if one can be transformed into the other via a deformation of $\mathbb{R}^3$ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.”

Professor Heath showed us several knot diagrams, explained three legal “Reidemeister moves” that can be used to transform a knot, and then described how to determine if two knots are the same using “colorability” of the strands of a knot diagram as a classification criterion. Then he presented an important theorem: “If we have an unknot in a diagram with $n$ crossings, then it can be unknotted in less than $2^{n \times 10^{11}}$ Reideimeister moves.” This was apparently a huge breakthrough in knot theory, because it showed that there is an upper bound on the number of moves necessary to unknot a knot, but it is such a large number, that it is relatively useless from a practical standpoint.

I found the presentation interesting and informative, and was actually able to follow most of it. I found myself becoming a little lost when a theorem was presented about “n-colorability.” It is defined using modular arithmetic, which I understand, but the application to the knot diagrams went a bit “fast” for me. Even though there are important practical applications to knot theory, such as the properties of knotted vs. unknotted chemicals in chemistry, and string theory in physics, I find the whole concept of knot theory somewhat unusual. I know that we were exposed to only a small portion of what knot theory entails, but at this point in time, I don’t think it is a topic I am interested in pursuing. I meant to ask Professor Heath after the lecture why he became interested in knot theory initially, and also why he has continued to study the topic, because I am always interested in what motivates people to do what they do. I guess that will have to wait for another day.