Ribbonlength of families of folded ribbon knots

Denne, Elizabeth; Haden, John Carr; Larsen, Troy; Meehan, Emily

doi:10.2140/involve.2022.15.591

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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

DOI: 10.2140/involve.2022.15.591

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 $2 p$ . 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 sublinear in crossing number.

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Keywords

knots, links, folded ribbon knots, ribbonlength, crossing number, 2-bridge knots, torus knots, pretzel knots, twist knots

Mathematical Subject Classification

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

Vol. 15, No. 4, 2022