Mathematics

Knot Theory Research Project

Knot theory is a branch of topology that involves the study of knotted geometric objects. A mathematical knot is a closed curve in three dimensional space that does not intersect itself. One central question in knot theory is can we determine whether two given knots are actually the same knot, in other words can one be deformed into the other without making any cuts? For example, the following knot diagrams represent the same knot and are known as the "Perko pair" [1]. Can you find a series of deformations that relate the pair?

Perko Pair

Knot theory has been formally studied since the 1880s. Mathematicians have studied knots for their interesting and beautiful geometric properties and for the applications of knot theory in the sciences. In the 1980s, biochemists discovered knotting in DNA molecules and mathematicians have been interested in how knot theory can be used in the study of DNA ever since.

In this research project we will explore the applications of knot theory to DNA. The main topic that we will study is a calculus of rational tangles that can be used to model the actions of enzymes on DNA. Enzymes modify DNA to perform various biological functions, but the actual actions of the enzymes on the DNA strands are hard to determine. For example, if the following diagram represents circular knotted DNA before and after the enzyme (in blue) modified its strands, what action did the enzyme actually perform?

Enzyme Reaction

We will start with an introduction to both knots and DNA, so prior study of these topics is not necessary. We will then study several recent papers on using knot theory to model DNA and look at some open questions. Students who are interested in both mathematics and biology will find that this project is a great way to explore parts of both fields. I expect that this summer research project will provide a number of opportunities for further research for students interested in continued study.

For more information about this research project contact Bill Schellhorn, Carver 331B.

[1] These knot diagrams are from the webpage http://www.knotplot.com/perko/

Last Updated: 11/18/11