NSF Award
0074315
The investigator studies connections between knot complexity<br/>and polygonal knot spaces, and develops effective methods to<br/>quantify and and characterize knots. The project involves<br/>computation and software development as well as analysis and<br/>experiments. With colleagues and students, the investigator<br/>explores relationships between various experimental measurements<br/>of complexity of knots in physical materials, such as DNA and<br/>polymers, and mathematical characterizations of knot complexity.<br/>Previously defined functions, such as energies and rope-length,<br/>are compared to new quantities, such as measurements of the<br/>convex hull and of a "smallest" box containing the knot, to<br/>capture various spatial characteristics. These quantities<br/>predict the types of knots that are encountered as one moves<br/>through knot space. They also are used to understand changes<br/>that occur in small and large-scale knotting in polygonal knot<br/>space as a result of perturbations.<br/> From DNA replication to unraveling one's garden hose,<br/>knotting and tangling are a part of many physical systems. Some<br/>knots are easier to tie (i.e. less complex), and thus more likely<br/>to occur in these situations. How does one quantify the<br/>complexity of a knot? What measurable attributes fully explain<br/>the complexity of a mathematical knot (i.e. a closed loop in<br/>space)? Mathematicians have defined several functions, called<br/>"knot energies" that quantify the "tangledness" of knots.<br/>Simultaneously, scientists have completed physical experiments on<br/>knots made of real materials, such as DNA and polymers, that<br/>determine other measures of complexity. To what extent are the<br/>theoretical and experimental quantities related? Are the<br/>quantities delivering the same information or does each number<br/>reveal something different about the knot? In particular, can<br/>one use these functions to create more realistic physical models<br/>of DNA? In this project, the investigator, colleagues, and<br/>students explore the quantification of knot complexity and its<br/>relation to spaces of polygonal knots by integrating theory with<br/>computer simulation. Previously defined theoretical measures,<br/>such as energies and rope-length, are compared to new quantities,<br/>such as the surface area and volume of the convex hull, to<br/>capture various spatial characteristics related to the knot.<br/>Physical experiments and computer simulations are performed and<br/>statistical analysis applied to understand their interrelations.<br/>These quantities also predict the types of knots that are<br/>encountered as one moves through polygonal knot space and<br/>explains changes that occur in small and large-scale knotting as<br/>a result of perturbations. This provides scientists with a<br/>better understanding of the mathematical models that are<br/>currently employed and suggest refinements to improve these<br/>models.
