The Geometry and Topology of Wide Ribbons

Abstract

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 R. Given an arclength parametrization of K, denoted by γ(s), and given a smooth, smoothly-closed, unit vector field u(s) with the property that u′(s) 6= 0 for any s in the domain, we may define a ribbon of generalized width r0 associated to γ and u as the set of all points γ(s) + ru(s) for all s in the domain and for all r ∈ [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 γ. However, the particular knot type in general depends on γ. The occurrence of stabilized knot types for large widths is generic; we show that the set of pairs (γ,u) for which the outer ribbon edge stabilizes for large widths (as a subset of all such pairs (γ,u)) is open and dense in the C topology. Finally, we provide an algorithm for constructing a