Ribbonlength of families of folded ribbon knots

TL;DR

This paper investigates the folded ribbonlength of various knot families, providing linear and sub-linear upper bounds related to crossing number, and introduces a new folding method for torus knots.

Contribution

It establishes new upper bounds on folded ribbonlength for multiple knot families, including a novel folding technique for $(p,q)$ torus knots.

Findings

Upper bounds are linear in crossing number for 2-bridge, torus, twist, and pretzel knots.

A new folding method for $(p,q)$ torus knots achieves an upper bound of $2p$.

Existence of a sub-linear upper bound proportional to the square root of crossing number for certain torus knots.

Abstract

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\geq q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot Cr(K)^{1/2}$. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.