Knots Connected by Wide Ribbons

Abstract

A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$ is a unit vector field based along $x$. In the 1960s and 1970s, G. Calugareanu, G. H. White, and F. B. Fuller proved relationships between the geometry and topology of thin ribbons, in particular the "Link = Twist + Writhe" theorem that has been applied to help understand properties of double-stranded DNA. Although ribbons of small width have been studied extensively, it appears that less is known about ribbons of large width whose images (even via a smooth map) can be singular or self-intersecting. Suppose $K$ is a smoothly embedded knot in $\mathbb{R}^3$. Given a regular parameterization $\mathbf{x}(s)$, and a smooth unit vector field $\mathbf{u}(s)$ based along $K$, we may define a ribbon of width $R$ associated to $\mathbf{x}$ and $\mathbf{u}$ as the set of all points $\mathbf{x}(s) + r\mathbf{u}(s)$, $r \in [0,R]$. For large $R$, these wide ribbons typically have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge $\mathbf{x}(s) + R\mathbf{u}(s)$ relates to that of the original knot $K$. We show that, generically, there is an eventual limiting knot type of the outer ribbon edge as $R$ gets arbitrary large. We prove that this eventual knot type is one of only finitely many possibilities which depend just on the vector field $\mathbf{u}$. However, the particular knot type within the finite set depends on the parameterized curves $\mathbf{x}(s)$, $\mathbf{u}(s)$, and their interactions. Finally, we show how to control the curves and their parameterizations so that given two knot types $K_1$ and $K_2$, we can find a smooth ribbon of constant width connecting curves of these two knot types.