Math Department (Thesis/2007-08)

Knot Theory

Under the direction of: Brenda Johnson

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Knot theory is the mathematical study of knots. As a mathematical discipline, knot theory had its origins in chemistry and physics. At one time, it was believed that atoms were composed of "knotted" matter, and that different elements were determined by different knots. As a result, it became very important to develop techniques for distinguishing between different types of knots.

Anyone who's ever tried to unknot a particularly messy knot has run into some of the same difficulties that make knot theory a challenging field of study. Over the past century, knot theorists have developed a wide range of techniques for distinguishing between and "unknotting" knots that circumvent the problems encountered in unknotting one's shoelaces. Many of these techniques are covered in the course in knot theory (Math 225) offered by the department. But there are many topics that are not covered in the course and many open problems in knot theory that make ideal topics for undergraduate research and theses. Among these topics and problems are the following:

Stick numbers. A "stick knot" is a knot constructed of rigid sticks, as opposed to more flexible string or rope. The "stick number" of a knot is the smallest number of sticks necessary to construct that knot. The stick numbers are known for a small collection of the simplest knots, but very little is known in general about stick numbers. Some estimates have been made that relate the stick number of a knot to the number of times a knot crosses over or under itself, but there is considerable room for improvement in these estimates. A thesis project in this area could involve determining the stick numbers in cases where the stick numbers are not known, or trying to develop more general techniques for estimating the stick number of a knot.

Khovanov homology. The introduction of the Jones Polynomial in 1984 revolutionized the field of knot theory. The Jones polynomial is a polynomial obtained from a picture of a knot by a particular algorithm. The algorithm is constructed in such a way that any picture of the knot will produce the same polynomial. (In mathspeak, this means that the Jones polynomial is a knot invariant.) In 2000, M. Khovanov introduced a more general invariant, now known as Khovanov homology, that gives you the Jones polynomial plus much more information. Khovanov homology is currently a hot topic in knot theory, causing a minor revolution of its own.

A thesis in this area would require learning some introductory graduate-level algebra, before learning about Khovanov homology. This topic would be most suitable for a two-term thesis or two-term honors thesis.

Prerequisites: For all Knot Theory projects, Math 225 is recommended, but not required. Math 332 is required for Khovanov homology.

Math Department web pages Created: 25 Apr 2007 Last modified: 25 Apr 2007 Comments to: math@union.edu

Thesis Topics - K. Plofker Thesis Topics for 2007-2008 Thesis Topics - B. Johnson