Ribbonlength of families of folded ribbon 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, p) torus, twist, and pretzel knots, and these upper bounds turn out to be linear in crossing number. We give a new way to fold (p, q) torus knots, and show that their folded ribbonlength is bounded above by p + q. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 5. We then show that any (p, q) torus knot K has a constant c > 0, such that the folded ribbonlength is bounded above by c • Cr(K) 1/2 , providing an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.

Ribbonlength of twisted torus knots

“…If r is equal to 1, then T p,q;r,s has the same knot type with T p,q regardless of the value of s. This implies that when we take the value of s as 1 in Theorem 4, we have the result that Rib(T p,q ) ≤ 2p. It is also found by Denne el al [4].…”

“…He used the pentagonal and the hexagonal folded ribbon knots. For torus knots, several researchers have given upper bounds of ribbonlength [4,8] In this paper, we deal with the ribbonlength of a specific type of knot called a twisted torus knot. A twisted torus knot T p,q;r,s is obtained from the torus knot T p,q by twisting r adjacent strands s full twists.…”