The Geometry and Topology of Wide Ribbons

Intuitively, a ribbon is a topological and geometric surface that has a fixed width. In the 1960s and 1970s, Calugareanu, White, and Fuller each independently proved a relationship between the geometry and topology of thin ribbons. This result has been applied in mathematical biology when analyzing properties of DNA strands. Although ribbons of small width have been studied extensively, it appears as though little to no research has be completed regarding ribbons of large width.

In general, suppose K is a smoothly embedded knot in R³. Given an arclength parametrization of K, denoted by gamma(s), and given a smooth, smoothly-closed, unit vector field u(s) with the property that u'(s) is not equal to 0 for any s in the domain, we may define a ribbon of generalized width r₀ associated to gamma and u as the set of all points gamma(s) + ru(s) for all s in the domain and for all r in [0,r₀]. These wide ribbons are likely to have self-intersections. In this thesis, we analyze how the knot type of the outer ribbon edge relates to that of the original knot K and the embedded resolutions of the unit vector field u as the width increases indefinitely. If the outer ribbon edge is embedded for large widths, we prove that the knot type of the outer ribbon edge is one of only finitely many possibilities. Furthermore, the possible set of finitely many knot types is completely determinable from u, independent of gamma. However, the particular knot type in general depends on gamma. The occurrence of stabilized knot types for large widths is generic; we show that the set of pairs (gamma, u) for which the outer ribbon edge stabilizes for large widths (as a subset of all such pairs (gamma, u) is open and dense in the C¹ topology. Finally, we provide an algorithm for constructing a ribbon of constant generalized width between any two given knot types K₁ and K₂. We conclude by providing concrete examples.