Ribbonlength and crossing number for folded ribbon knots

We study Kauffman's model of folded ribbon knots: knots made a thin strip paper flat in the plane. The ribbonlength is length to width ratio such knot. show for any knot or link type that there exist constants $c_1, c_2>0$ bounded above by $c_1\cdot Cr(K)^2$, and also $c_2\cdot Cr(K)^{3/2}$. use different method each bound. constant $c_1$ quite small comparison $c_2$, first bound lower than second links with $Cr(K)\leq$ 12,748.