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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

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 q > 2 has a constant c > 0, such that the folded ribbonlength is bounded above by c Cr (K)12. This provides an example of an upper bound on folded ribbonlength that is sublinear in crossing number.

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

Communicated by Joel Foisy
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