Ribbonlength upper bounds for small crossing knots and links

Abstract

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. By finding new methods of creating folded ribbon knots, we improve upon existing upper bounds for the folded ribbonlength of $(2,q)$-torus links, twist knots, and pretzel links. These give the best known bounds to date for small crossing knots in these families. For example, there is a folded ribbonlength twist knot $T_n$ with folded ribbonlength $\text{Rib}(T_n) = n +6$. Applying this to the figure-eight knot $T_2$ yields a folded ribbonlength $\text{Rib}(T_2)= 8$, which we conjecture is the infimum.