Ribbonlength of families of folded ribbon knots

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 >= q > 2 has a constant c > 0, such that the folded ribbonlength is bounded above by c center dot Cr(K)1/2. This provides an example of an upper bound on folded ribbonlength that is sublinear in crossing number.