Ribbonlength of torus knots

Abstract

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised 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)) (resp., 2q cot(\pi/(2q+1))). Using these calculations, we provide the bounds c_1 \leq 2/\pi and c_2 \geq 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) \leq R(K) \leq c_2 C(K).