Knot Complexity and the Structure of Polygonal Knot Space

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The investigator studies connections between knot complexityand polygonal knot spaces, and develops effective methods toquantify and and characterize knots. The project involvescomputation and software development as well as analysis andexperiments. With colleagues and students, the investigatorexplores relationships between various experimental measurementsof complexity of knots in physical materials, such as DNA andpolymers, and mathematical characterizations of knot complexity.Previously defined functions, such as energies and rope-length,are compared to new quantities, such as measurements of theconvex hull and of a "smallest" box containing the knot, tocapture various spatial characteristics. These quantitiespredict the types of knots that are encountered as one movesthrough knot space. They also are used to understand changesthat occur in small and large-scale knotting in polygonal knotspace as a result of perturbations. From DNA replication to unraveling one's garden hose,knotting and tangling are a part of many physical systems. Someknots are easier to tie (i.e. less complex), and thus more likelyto occur in these situations. How does one quantify thecomplexity of a knot? What measurable attributes fully explainthe complexity of a mathematical knot (i.e. a closed loop inspace)? Mathematicians have defined several functions, called"knot energies" that quantify the "tangledness" of knots.Simultaneously, scientists have completed physical experiments onknots made of real materials, such as DNA and polymers, thatdetermine other measures of complexity. To what extent are thetheoretical and experimental quantities related? Are thequantities delivering the same information or does each numberreveal something different about the knot? In particular, canone use these functions to create more realistic physical modelsof DNA? In this project, the investigator, colleagues, andstudents explore the quantification of knot complexity and itsrelation to spaces of polygonal knots by integrating theory withcomputer 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, tocapture various spatial characteristics related to the knot.Physical experiments and computer simulations are performed andstatistical analysis applied to understand their interrelations.These quantities also predict the types of knots that areencountered as one moves through polygonal knot space andexplains changes that occur in small and large-scale knotting asa result of perturbations. This provides scientists with abetter understanding of the mathematical models that arecurrently employed and suggest refinements to improve thesemodels.