St. Mary's College

Titles:

Director: Katherine Socha

Dates of Programs: June 16 - July 25, 2008

Summaries:

(a) In the 1980s, Vaughn Jones revolutionized the field of Knot Theory with the discovery of a new method of determining knot equivalence. The Jones Polynomial, as the method is known, has been widely studied since then, yet continues to offer new avenues for research at all levels. One of the most basic questions remains unanswered: does every nontrivial knot have a nontrivial Jones Polynomial? A new and elementary technique, developed by Ganzell, may shed some light on this problem. In this project we will study various families of knots and try to show that no such nontrivial knot exists.
(b) The students will investigate discrete curves and discrete surfaces in Euclidean three space. The project will include paper and pencil, computer visualization, and constructing curves with plastic and metal. The ability to change the angle between segments allows change in the discrete curvature of a curve. The ability to twist the caps at the ends of the segments allows change in the discrete torsion of a curve. Finally, the ability to attach four (or more) segments allow the construction of discrete surfaces. Comparisons with smooth curves and surfaces, both in the local and global theories, will be studied. As time permits mathematical models for DNA twisted pairs will be considered in this context.

Student Researchers Supported by MAA: