Ribbonlength of torus knots

Using Kauffman's model of. at knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1) cot(pi/(2q + 1)) (respectively, 2q cot(pi/(2q + 1))). Using these calculations, we provide the bounds c(1) <= 2/pi and c(2) >= 5/3 cot pi/5 for the constants c(1) and c(2) that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c(1)C(K) <= R(K) <= c(2)C(K).