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Ribbonlength of families of folded ribbon knots

### Elizabeth Denne, John Carr Haden, Troy Larsen and Emily Meehan

Vol. 15 (2022), No. 4, 591–628
##### 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, $\left(2,q\right)$ 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 $\left(p,q\right)$ 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 $\left(p,q\right)$ torus knot $K$ with $p\ge q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot \mathrm{Cr}{\left(K\right)}^{1∕2}$. This provides an example of an upper bound on folded ribbonlength that is sublinear in crossing number.

##### Keywords
knots, links, folded ribbon knots, ribbonlength, crossing number, 2-bridge knots, torus knots, pretzel knots, twist knots
Primary: 57K10
##### Milestones
Received: 9 November 2020
Revised: 23 July 2021
Accepted: 12 January 2022
Published: 7 January 2023

Communicated by Joel Foisy
##### Authors
 Elizabeth Denne Department of Mathematics Washington & Lee University Lexington, VA United States John Carr Haden Department of Mathematics Washington & Lee University Lexinton, VA United States Troy Larsen Department of Mathematics Washington & Lee University Lexington, VA United States Emily Meehan Department of Mathematics Wheaton College Norton, MA United States