2020/10/08 by Denne, Elizabeth, Haden, John Carr, Larsen, Troy +1 · 3 citations
#57K10 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2010.04188
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)1/2. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.