Ribbonlength and crossing number for folded ribbon knots

TL;DR

This paper investigates bounds on the ribbonlength of folded ribbon knots, establishing quadratic and sub-quadratic upper bounds related to crossing number, with implications for understanding the geometry of folded knots.

Contribution

It introduces new bounds on ribbonlength in terms of crossing number, using different methods for each bound, advancing the theoretical understanding of folded ribbon knots.

Findings

Ribbonlength is bounded above by a quadratic function of crossing number.

Another bound shows ribbonlength is also limited by a 3/2 power of crossing number.

For knots with crossing number up to 12,748, the quadratic bound is tighter.

Abstract

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants $c_1, c_2>0$ such that the ribbonlength is bounded above by $c_1\cdot Cr(K)^2$, and also by $c_2\cdot Cr(K)^{3/2}$. We use a different method for each bound. The constant $c_1$ is quite small in comparison to $c_2$, and the first bound is lower than the second for knots and links with $Cr(K)\leq$ 12,748.