Determining if two knots are not equivalent in an efficient manner is important in the study of knots. The arrow polynomial, which is calculated from a virtual knot diagram and is invariant under the Reidemeister moves, can be used to determine if two knots are not equivalent and determine a lower bound on the virtual crossing number. In this paper, we present the necessary data structures and algorithms to represent a link diagram on a computer and calculate the arrow polynomial.

Author Bio

I am a senior at McKendree University in the Department of Mathematics. In the fall semester of 2008, I was enrolled in the senior seminar course in Mathematics whose one major requirement is the completion of an independent research project. I chose to work with Dr. Heather Dye, a knot theorist, and designed data structure to represent oriented knots on a computer, and then developed algorithm to reduce a virtual knot and compute the arrow polynomial. I found the topic highly engaging. I continually worked with Dr. Dye, even after the course was completed, to compute the arrow polynomial of different types of knots. In the future, I plan to pursue a doctoral degree in the field of combinatorics and make a career in research.