Knot Complexity and the Structure of Polygonal Knot Space

Chatham College, Pittsburgh PA

Investigators

Abstract

The investigator studies connections between knot complexity and polygonal knot spaces, and develops effective methods to quantify and and characterize knots. The project involves computation and software development as well as analysis and experiments. With colleagues and students, the investigator explores relationships between various experimental measurements of complexity of knots in physical materials, such as DNA and polymers, and mathematical characterizations of knot complexity. Previously defined functions, such as energies and rope-length, are compared to new quantities, such as measurements of the convex hull and of a "smallest" box containing the knot, to capture various spatial characteristics. These quantities predict the types of knots that are encountered as one moves through knot space. They also are used to understand changes that occur in small and large-scale knotting in polygonal knot space as a result of perturbations. From DNA replication to unraveling one's garden hose, knotting and tangling are a part of many physical systems. Some knots are easier to tie (i.e. less complex), and thus more likely to occur in these situations. How does one quantify the complexity of a knot? What measurable attributes fully explain the complexity of a mathematical knot (i.e. a closed loop in space)? Mathematicians have defined several functions, called "knot energies" that quantify the "tangledness" of knots. Simultaneously, scientists have completed physical experiments on knots made of real materials, such as DNA and polymers, that determine other measures of complexity. To what extent are the theoretical and experimental quantities related? Are the quantities delivering the same information or does each number reveal something different about the knot? In particular, can one use these functions to create more realistic physical models of DNA? In this project, the investigator, colleagues, and students explore the quantification of knot complexity and its relation to spaces of polygonal knots by integrating theory with computer simulation. Previously defined theoretical measures, such as energies and rope-length, are compared to new quantities, such as the surface area and volume of the convex hull, to capture various spatial characteristics related to the knot. Physical experiments and computer simulations are performed and statistical analysis applied to understand their interrelations. These quantities also predict the types of knots that are encountered as one moves through polygonal knot space and explains changes that occur in small and large-scale knotting as a result of perturbations. This provides scientists with a better understanding of the mathematical models that are currently employed and suggest refinements to improve these models.